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MATICS. By Augustus Db Morgan. Entirely new edi- 
tion, with portrait of the author, index, and ^nnotationsi 
bibliographies of modern works on algebra, the philosophy 
of mathematics, pan-geometry, etc. Pp., 288. Cloth, $1.25 
net (48. 6d. net). 

Joseph Louis Lagrange. Translated from the French by 
Thomas J. McCormack. With photogravure portrait of 
Lagrange, notes, biography, marginal analyses, etc. Only 
separate edition in French or English. Pages, 172. Cloth, 
f i.oo net (48. 6d. net). 

Morgan. New reprint edition. With sub-headings, and 
a brief bibliography of English, French, and Germaif text- 
books of the Calculus. Pp., 144. Price, fi.oo;(4s. 6d. net). 

Hermann Schubert, Professor of Mathematics in the 
Johanneum, Hamburg, Germany. Translated from the 
German by Thomas J. McCormack. Containing essays on 
The Notion and Definition of Number, Monism in Arith- 
metic, The Nature of Mathematical Knowledge, The 
Magic Square, The Fourth Dimension, The Squaring of 
the Circle. Pages, 149. Cuts, 37. Price, Cloth, 75c net 
(3s. net). 

By Dr. Karl Fink, late Professor in Tiibingen. Translated 
from the German by Prof. Wooster Woodruff Beman and 
Prof. David Eugene Smith. Pp. 333. Price, cloth, fi.50 
net (58. 6d. net). 


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n^HB translators feel that oo apology is necessary for any rea- 
^ sonable effort to encourage the study of the history of mathe- 
matics. The clearer view of the science thus afforded the teacher, 
the inspiration to improve his methods of presenting it, the in- 
creased interest in the class-work, the tendency of the subject to 
combat stagnation of curricula, — these are a few of the reasons for 
approving the present renaissance of the study. 

This phase of scientific history which Montucla brought into 
such repute — it must be confessed rather by his literary style than 
by his exactness — and which writers like De Morgan in England, 
Chasles in France, Quetelet in Belgium, Hankel and Baltzer in 
Germany, and Boncompagni in Italy encouraged as the century 
wore on, is seeing a great revival in our day. This new movement 
is headed by such scholars as Gtinther, Enestrom, Loria, Paul 
Tannery, and Zeuthen. but especially by Moritz Cantor, whose 
VorUsunffen Uber Geschichte der Mathemaiik must long remain 
the world's standard. 

In any movement of this kind compendia are always necessary 
for those who lack either the time or the linguistic power to read 
the leading treatises. Several such works have recently appeared 
in various languages. But the most systematic attempt in this 
direction is the work here translated. The writers of most hand- 
books of this kind feel called upon to collect a store of anecdotes, 
to incorporate tales of no historic value, and to minimize the real 
history of the science. Fink, on the other hand, omits biography 
entirely, referring the reader to a brief table in the appendix or to 
the encyclopedias. He systematically considers the growth of 


arithmetic, algebra, geometry, and trigonometry, carrying the his- 
toric development, as should be done, somewhat beyond the lipiits 
of the ordinary course. 

At the best, the work of the translator is a rather thankless 
task. It is a target for critics of style and for critics of matter. 
For the style of the German work the translators will hardly be 
held responsible. It is not a fluent one, leaning too much to the 
scientific side to make it always easy reading. Were the work 
less scientific, it would lend itself more readily to a better English 
form, but the translators have preferred to err on the side of a 
rather strict adherence to the original. 

As to the matter, it has seemed unwise to make any consider- 
able changes. The attempt has been made to correct a number of 
unquestionable errors, occasional references have been added, and 
the biographical notes h^ve been rewritten. It has not seemed 
advisable, however, to insert a large number of bibliographical 
notes. Readers who are interested in the subject will naturally 
place upon their shelves the works of De Morgan, Allman, Gow, 
Ball, Heath, and other English writers, and, as far as may be, 
works in other languages. The leading German authorities are 
mentioned in the footnotes, and the French language offers little 
at present beyond the works of Chasles and Paul Tannery. 

The translators desire to express their obligations to Professor 
Markley for valuable assistance in the translation 

Inasmuch as the original title of the work, Geschichte der 
EUmentar-Mathematik, is misleading, at least to English read- 
ers, the work going considerably beyond the limits of the elements, 
it has been thought best to use as the English title, A Brief His- 
tory of Mathematics. 

W. W. Beman, Ann Arbor, Mich. 
D. E. Smith, New York. 
March, 1900. 





TF the history of a science possesses value for every one whom 
-*- calling or inclination brings into closer relations to it, — if the 
knowledge of this history is imperative for all who have influence 
in the further development of scientific principles or the methods 
of employing them to advantage, then acquaintance with the rise 
and growth of a branch of science is especially important to the 
man who wishes to teach the elements of this science or to pene- 
trate as a student into its higher realms. 

The following history of elementary mathematics is intended 
to give students of mathematics an historical survey of the ele- 
mentary parts of this science and to furnish the teacher of the ele- 
ments opportunity, with little expenditure of time, to review con- 
nectedly points for the most part long familiar to him and to utilise 
them in his teaching in suitable comments. The enlivening in- 
fluence of historical remarks upon this elementary instruction has 
never been disputed. Indeed there are text-books for the elements 
of mathematics (among the more recent those of Baltzer and Schu- 
bert) which devote considerable space to the history of the science 
in the way of special notes. It is certainly desirable that instead 
of scattered historical references there should be offered a con- 
nected presentation of the history of elementary mathematics, not 
one intended for the use of scholars, not as an equivalent for the 
great works upon the history of mathematics, but only as a first 
picture, with fundamental tones clearly sustained, of the principal 
results of the investigation of mathematical history. 

In this book the attempt has been made to differentiate the 
histories of the separate branches of mathematical science. There 


are considered in order number-systems and number-symbols, 
arithmetic, algebra, geometry and trigonometry, allowing, as far 
as possible, within the narrow confines of a single branch of the 
elements, a rapid and sure orientation. Against such a procedure 
the objection may be raised that in this way the general survey of 
the culture history of a certain epoch will suffer. On the other 
hand, in a history of elementary mathematics, especially one con- 
fined within such modest bounds, an exhaustive description of 
whole periods with all their correlations of past and future cannot 
well be presented. 

It is not the purpose of this work to set forth the interesting 
historical development of mechanics and astronomy. Although it 
cannot be denied that by this separation of related branches there 
is wanting a certain definitiveness to the work, yet the hope may 
be expressed that this lack will not be felt too keenly. The ele- 
mentary parts of mathematics have only few points of contact with 
these branches, and our endeavor is to present in brief compass 
only that which is most essential. 

Further, in the interest of a presentation as condensed as pos- 
sible, the biographical notices which often lend great attraction 
to a more extended treatment of a subject must be relegated to the 
appendix and there treated but briefly. 

The work had its inception in certain suggestions which the 
author received at the semi-monthly meetings of a mathematical 
club in Tubingen, founded and conducted by Prof. Dr. A. Brill, 
for which suitable thanks ought here to be expressed. Acknowl- 
edgment is especially due to the president of the club whose in- 
terpretations have been decisive for certain parts of the present 
work. These meetings furnished the author the desired oppor- 
tunity, through the lectures connected with the most diverse 
branches of the science and through the discussions which often 
followed, with references to recent literature, to penetrate into 
those circles of thought which to-day dominate the higher branches 
of mathematics. The writer was thus led to complete his studies 


by going into the recent history of the science. T)ie results of 
such investigations are here presented with perhaps greater full- 
ness than seems necessary for the main purpose of the book or 
justified by its title. But in default of such a digest, a first experi- 
ment may lay claim to a friendly judgment, in spite of the con- 
tinually increasing subdivisions of the science ; nor will such an 
attempt be thought inappropriate, inasmuch as it does not seem 
possible to draw a sharp line of demarcation between the elemen- 
tary and higher mathematics. For on the one hand certain prob- 
lems of elementary mathematics have from time to time furnished 
the occasion for the development of higher branches, and on the 
other from the acquisitions of these new branches a clear light has 
fallen upon the elementary parts. Accordingly it may be gratify- 
ing to many a student and teacher to find here at least that which 
is fundamental. 

The exceedingly rich literature, especially in German, at the 
disposal of the author is referred to in the footnotes. He has made 
free use of the excellent Jahrbuch Hber die Fartschrilte der 
McUhemcUik, which with clear and systematic arrangement enu- 
merates and discusses the most recent mathematical literature. 

K. Fink 
Tt)BiNGEN. June. 1890. 



Translators' Preface iii 

Author's Preface v 

General Survey i 



A. General Survey i8 

B. First Period. The Arithmetic of the Oldest Nations to 

the Time of the Arabs. 

1. The Arithmetic of Whole Numbers 24 

2. The Arithmetic of Fractions 31 

3. Applied Arithmetic 34 

C. Second Period. From the Eighth to the Fourteenth Cen- 


1. The Arithmetic of Whole Numbers 36 

2. The Arithmetic of Fractions 40 

3. Applied Arithmetic .... 41 

D. Third Period. From the Fifteenth to the Nineteenth Cen- 


1. The Arithmetic of Whole Numbers 41 

2. The Arithmetic of Fractions 49 

3. Applied Arithmetic 51 


A. General Survey 61 

B. First Period. From the Earliest Times to the Arabs. 

I. General Arithmetic 63 

Egyptian Symbolism 63. Greek Arithmetic 64; Symbolism 
6s; Theory of Numbers 66; Series 67; the Irrational 68; Neg- 



ative Numbers 70; Archimedes's Notation for Large Numbers 
71. Roman Arithmetic 71. Hindu Arithmetic 71 ; Symbolism 
72; Negative Numbers 72; Involution and Evolution 73; Per- 
mutations and Combinations 74; Series 74. Chinese Arith- 
metic 74. Arab Arithmetic 74 ; "Algorism " 75 ; Radical Signs 
76 ; Theory of Numbers 76 ; Series 76. 

2. Algebra . . ^ 77 

The Egyptians 77. The Greeks; Form of the Equation 77; 
Equations of the First Degree 78; Equations of the Second 
Degree (Application of Areas) 79; Equations of the Third De- 
gree 81 ; Indeterminate Equations (Cattle Problem of Archi- 
medes; Methods of Solution of Diophantus) 83. Hindu Al- 
gebra 84. Chinese Algebra 87. Arab Algebra 88. 

C. Second Period. To the Middle of the Seventeeli',h Cen- 


1. Greneral Arithmetic 95 

Symbolism of the Italians and the German Cossists 95 ; Irra- 
tional and Negative Numbers 99; Imaginary Quantities xoi ; 
Powers 102; Series t03 ; Stifel's Duplication of the Cube 104 ; 
Magic Squares 105. 

2. Algebra 107 

Representation of Equations 107; Equations of the First and 
Second Degrees 108; Complete Solution of Equations of the 
Third and Fourth Degrees by the Italians iii ; Work of the 
German Cossists 113 ; Beginnings of a General Theory of Al- 
gebraic Equations 115. 

D. Third Period. From the Middle of the Seventeenth Cen- 

tury to the Present Time. 

Symbolism 117; Pascal's Arithmetic Triangle 118; Irrational 
Numbers 119; Complex Numbers 123; Grassmann's Aut- 
dehnungslehre 127 ; Quaternions 129 ; Calculus of Logic 131 ; 
Continued Fractions 131 ; Theory of Numbers 133; Tables of 
Primes 141; Symmetric Functions 142; Elimination 143; The- 
ory of Invariants and Covariants 145 ; Theory of Probabilities 
148 ; Method of Least Squares 149 ; Theory of Combinations 
150; Infinite Series (Convergence and Divergence) 151 ; Solu- 
tion of Algebraic Equations 155; the Cyclotomic Equation 
160; Investigations of Abel and Galois 163; Theory of Substi- 
tutions 164; the Equation of the Fifth Degree 165; Approxi- 
mation of Real Roots 166 ; Determinants 107; Differential and 
Integral Calculus 168; Differential Equations 174; Calculus 
of Variations 178 ; Elliptic Functions 180; Abelian Functions 
186; More Rigorous Ten iency of Analysis 189. 




A. General Survey 190 

B. First Period. Egyptians and Babylonians 192 

C. Second Period. The Greeks 193 

The Geometry of Thales and Pythagoras 194 ; Application of 
the Qaadratrix to the Quadrature of the Circle and the Trisec- 
tion of an Angle 196 ; the EUment* of Euclid 198 ; Archimedes 
and his Successors 199 ; the Theory of Conic Sections 20a ; 
the Duplication of the Cube, the Trisection of an Angle and 
the Quadrature of the Circle 209; Plane, Solid, and Linear 
Loci 209; Surfaces of the Second Order 312; the Stereo- 
graphic Projection of Hipparchus 213. 

D. Third Period. Romans, Hindus, Chinese, Arabs . . . 214 

E. Fourth Period. From Gerbert to Descartes 218 

Gerbert and Leonardo 218; Widmann and Stifel22o; Vieta 
and Kepler 222; Solution of Problems with but One Opening 
of the Compasses 225; Methods of Projection 326. 

F. Fifth Period. From Descartes to the Present .... 228 

Descartes's Analytic Geometry 230; Cavalieri's Method of In- 
divisibles 334 ; Pascal's Geometric Works 237; Newton's In- 
vestigations 239; Cramer's Paradox 240; Pascal's Lima9on 
and other Curves 241 ; Analytic Geometry of Three Dimen- 
sions 242; Minor Investigations 243; Introduction of Projec- 
tive Geometry 246 ; lAHlbxns' s Barycentrtscher CalcBl 250 ; Bel- 
Imvitis's Equipollences 250; Plticker's Investigations 251; 
Steiner's Developments 256; Malfatti's Problem 256; Von 
Staudt's Geometrie <ier Lagt 258 ; Descriptive Geometry 359 ; 
Form-theory and Deficiency of an Algebraic Curve 261 ; 
Gauche Carves 263 ; Enumerative Geometry 264 ; Conformal 
Representation 266 ; Differential Geometry (Theory of Curva- 
ture of Surfaces) 267; Non-Euclidean Geometry 270 ; Pseudo- 
Spheres 273 ; Geometry of n Dimensions 275 ; Geomttria and 
Analysis Situs 275; Contact-transformations 276; Geometric 
Theory of Probability 276; Geometric Models 277; the Math- 
ematics of To-day 279. 


A. General Survey 281 

B. First Period. From the Most Ancient Times to the Arabs 282 

The Egyptians 382. The Greeks 282. The Hindus 284 The 
Arabs 385. 

• • 



C Second Period. From the Middle Ages to the Middle of 

the Seventeenth Century . . 287 

Vieta and Regiomontanas §87; Tkigonometrle Tables 389; 
Logarithms 390. 

D. Third Period From the Middle of the Seventeenth Cen- 
tury to the Present 294 

Biographical Notes 897 

Index , 323 



npHE beginnings of the development of mathemat- 
^ ical truths date back to the earliest civilizations 
of which any literary remains have come down to us, 
.namely the Egyptian and the Babylonian. On the 
one hand, brought about by the demands of practical 
life, on the other springing from the real scientific 
spirit of separate groups of men, especially of the 
priestly caste, arithmetic and geometric notions came 
into being. Rarely, however, was this knowledge 
transmitted through writing, so that of the Babylo- 
nian civilization we possess only a few traces. From 
the ancient Egyptian, however, we have at least one 
manual, that of Ahmes, which in all probability ap- 
peared nearly two thousand years before Christ. 

The real development of mathematical knowledge, 
obviously stimulated by Egyptian and Babylonian in- 
fluences, begins in Greece. • This development shows 
itself predominantly in the realm of geometry, and 
enters upon its first classic period, a period of no 
great duration, during the era of Euclid, Archimedes, 
Eratosthenes, and Apollonius. Subsequently it in- 
clines more toward the arithmetic side ; but it soon 
becomes so completely engulfed by the heavy waves 


of stormy periods that only after long centuries and 
in a foreign soil, out of Greek works which had es- 
caped the general destruction, could a seed, new and 
full of promise, take root. 

One would naturally expect to find the Romans 
entering with eagerness upon the rich intellectual 
inheritance which came to them from the conquered 
Greeks, and to find their sons, who so willingly re- 
sorted to Hellenic masters, showing an enthusiasm 
for Greek mathematics. Of this, however, we have 
scarcely any evidence. The Romans understood very 
well the practical value to the statesman of Greek 
geometry and surveying — a thing which shows itself 
also in the later Greek schools — but no real mathe- 
matical advance is to be found anywhere in Roman 
history. Indeed, the Romans often had so mistaken 
an idea of Greek learning that not infrequently they 
handed it down to later generations in a form entirely 

More important for the further development of 
mathematics are the relations of the Greek teachings 
to the investigations of the Hindus and the Arabs. 
The Hindus distinguished themselves by a pronounced 
talent for numerical calculation. What especially dis- 
tinguishes th'em is their susceptibility to the influence 
of Western science, the Babylonian and especially 
the Greek, so that they incorporated into their own 
system what they received from outside sources and 
then worked out independent results. 


The Arabs, however, in general do not show this 
same independence of apprehension and of judgment. 
Their chief merit, none the less a real one however, 
lies in the untiring industry which they showed in 
translating into their own language the literary treas- 
ures of the Hindus, Persians and Greeks. The courts 
of the Mohammedan princes from the ninth to the 
thirteenth centuries were the seats of a remarkable 
scientific activity, and to this circumstance alone do 
we owe it that after a period of long and dense dark- 
ness Western Europe was in a comparatively short 
time opened up to the mathematical sciences. 

The learning of the cloisters in the earlier part 
of the Middle Ages was not by nature adapted to 
enter seriously into matters mathematical or to search 
for trustworthy sources of such knowledge. It was 
the Italian merchants whose practical turn and easy 
adaptability first found, in their commercial relations 
with Mohammedan West Africa and Southern Spain, 
abundant use for the common calculations of arith- 
metic. Nor was it long after that there developed 
among them a real spirit of discovery, and the first 
great triumph of the newly revived science was the 
solution of the cubic equation by Tartaglia. It should 
be said, however, that the later cloister cult labored 
zealously to extend the Western Arab learning by 
means of translations into the Latin. 

In the fifteenth century, in the persons of Peur- 
bach and Regiomontanus, Germany first took position 


in the great rivalry for the advancement of mathemat- 
ics. From that time until the middle of the seven- 
teenth century the German mathematicians were 
chiefly calculators, that is teachers in the reckoning 
schools {Rechenschulen). Others, however, were alge- 
braists, and the fact is deserving of emphasis that 
there were intellects striving to reach still loftier 
heights. Among them Kepler stands forth pre-emi- 
nent, but with him are associated Stifel, Rudolff, and 
Biirgi. Certain is it that at this time and on Ger- 
man soil elementary arithmetic and common algebra, 
vitally influenced by the Italian school, attained a 
standing very conducive to subsequent progress. 

The modern period in the history of mathematics 
begins about the middle of the seventeenth century. 
Descartes projects the foundation theory of the ana- 
lytic geometry. Leibnitz and Newton appear as the 
discoverers of the differential calculus. The time has 
now come when geometry, a science only rarely, and 
even then but imperfectly, appreciated after its ban- 
ishment from Greece, enters along with analysis upon 
a period of prosperous advance, and takes full advan- 
tage of this latter sister science in attaining its results. 
Thus there were periods in which geometry was able 
through its brilliant discoveries to cast analysis, tem- 
porarily at least, into the shade. 

The unprecedented activity of the great Gauss 
divides the modern period into two parts : before 
Gauss — the establishment of the methods of the dif- 


ferential and integral calculus and of anal3rtic geom- 
etry as well as more restricted preparations for later 
advance ; with Gauss and after him — the magnificent 
development of modem mathematics with its special 
regions of grandeur and depth previously undreamed 
of. The mathematicians of the nineteenth century 
are devoting themselves to the theory of numbers, 
modern algebra, the theory of functions and projec- 
tive geometry, and in obedience to the impulse of 
human knowledge are endeavoring to carry their light 
into remote realms which till now have remained in 


AN inexhaustible profusion of external influences 
-^^ upon the human mind has found its legitimate 
expression in the formation of speech and writing 
in numbers and number-symbols. It is true that a 
counting of a certain kind is found among peoples of 
a low grade of civilization and even among the lower 
animals. "Even ducks can count their young."* But 
where the nature and the condition of the objects 
have been of no consequence in the formation of the 
number itself, there human counting has first begun. 
The oldest counting was even in its origin a pro- 
cess of reckoning, an adjoining, possibly also in special 
elementary cases a multiplication, performed upon 
the objects counted or upon other objects easily em- 
ployed, such as pebbles, shells, fingers. Hence arose 
number-names. The most common of these undoubt- 
edly belong to the primitive domain of language ; with 
the advancing development of language their aggre- 
gate was gradually enlarged, the legitimate combina- 

*Hankel, Zwr Geschichte der Mathematik im AUertmm und MitUlaUer, 
1874, p. 7. Hereafter referred to as Hankel. Tylor's Primitive Culture also 
has a valaable chapter apon counting. 


tion of single terms permitting and favoring the crea- 
tion of new numbers. Hence arose number-systems. 

The explanation of the fact that 10 is almost everyX 
where found as the base of the system of counting is j 
seen in the common use of the fingers in elementary 
calculations. In all ancient civilizations finger-reckon- 
ing was known and even to-day it is carried on to a 
remarkable extent among many savage peoples. Cer- 
tain South African races use three persons for num- 
bers which run above 100, the first counting the units 
on his fingers, the second the tens, and the third the 
hundreds. They always begin with the little finger of 
the left hand and count to the little finger of the right. 
The first counts continuously, the others raising a 
finger every time a ten or a hundred is reached.* 

Some languages contain words belonging funda- 
mentally to the scale of 5 or 20 without these systems 
having been completely elaborated ; only in certain 
places do they burst the bounds of the decimal sys- 
tem. In other cases, answering to special needs, 12 
and 60 appear as bases. The New Zealanders have^x 
a scale of 11, their language possessing words for the 
first few powers of 11, and consequently 12 is repre- 
sented as 11 and 1, 13 as 11 and 2, 22 as two ll's, 
and so on.f 

♦Cantor, M., Vorlttungen Uder Geschichte der Mathematik. Vol. I, 1880; 
2nd ed., 1894. p. 6. Hereafter referred to as Cantor. Conant, L. L., The Num- 
htr Omctpt, N. Y. 1896. Gow, J., History of Greek Geometry, Cambridge, 1884, 
Chap. 1. 

t Cantor, I., p. 10. 


In the verbal formation of a number-system addi- 
tion and multiplication stand out prominently as defin- 
itive operations for the composition of numbers ; very 
rarely does subtraction come into use and still more 
rarely division. For example, 18 is called in Latin 
10 + 8 {decern et octd)^ in Greek 8+10 (^Kr<i>-Kai-8cjca), 
in French 10 8 {dix-huif), in German 8 10 {acht-zehn), 
in Latin also 20 — 2 {duo-de-viginti^^ in Lower Breton 
3 6 {tri-omc*K)^ in Welsh 2-9 {dew-naw)^ in Aztec 
15 + 3 {caxtulli-otn-ey)y while 50 is called in the Basque 
half -hundred, in Danish two- and-a- half times twenty.* 
In spite of the greatest diversity of forms, the written 
representation of numbers, when not confined to th^ 
mere rudiments, shows a general law according t< 
which the higher order precedes the lower in the di; 
rection of the writing, f Thus in a four-figure number 
the thousands are written by the Phoenicians at the 
right, by the Chinese above, the former writing from 
right to left, the latter from above downward. A 
striking exception to this law is seen in the sub 
tractive principle of the Romans in IV, IX, XL, 
etc., where the smaller number is written before the 

Among the Egyptians we have numbers running 
from right to left in the hieratic writing, with varying 
direction in the hieroglyphics. In the latter the num- 
bers were either written out in words or represented 
by symbols for each unit, repeated as often as neces- 

*Hankel, p. 22. tHankel, p. 3a. 



sary. In one of the tombs near the pyramids of Gizeh 
have been found hieroglyphic numerals in which 1 is 
represented by a vertical line, 10 by a kind of horse- 
shoe, 100 by a short spiral, 10 000 by a pointing finger, 
100 000 by a frog, 1 000 000 by a man in the attitude 
of astonishment. In the hieratic symbols the figure 
for the unit of higher order stands to the right of the 
one of lower order in accordance with the law of se- 
quence already mentioned. The repetition of sym- 
bols for a unit of any particular order does not obtain, 
because there are special characters for all nine units, 
all the tens, all the hundreds, and all the thousands.* 
We give below a few characteristic specimens of the 
hieratic symbols : 

I LI III - -I A A *A - 

19 8 4 5 10 20 80 40 

The Babylonian cuneiform inscriptionsf proceed 
from left to right, which must be looked upon as ex- 
ceptional in a Semitic language. In accordance with 
the law of sequence the units of higher order stand on 
the left of those of lower order. The symbols used 
in writing are chiefly the horizontal wedge ^, the ver- 
tical wedge y, and the combination of the two at an 


angle ^. The symbols were written beside one another, 
or, for ease of reading and to save space, over one 
another. The symbols for 1, 4, 10, 100, 14, 400, re- 
spectively, are as follows : 

♦ Cantor, I., pp. 43, 44. t Cantor, I., pp. 77, 78. 


V VVV V W w VWV vvvw 

1 4 10 100 14 400 

For numbers exceeding 100 there was also, besides 
the mere juxtaposition, a multiplicative principle ; 
the symbol representing the number of hundreds was 
placed at the left of the symbol for hundreds as in the 
case of 400 already shown. The Babylonians probably 
had no symbol for zero.* The sexagesimal system 
(i. e., with the base 60), which played such a part in 
the writings of the Babylonian scholars (astronomers, 
and mathematicians), will be mentioned later. \ 

The Phoenicians, whose twenty-two letters werei 
derived from the hieratic characters of the Egyptians, ! 
either wrote the numbers out in words or used special ■ 
numerical symbols — for the units vertical marks, for/ 
the tens horizontal, f Somewhat later the Syrians used 
the twenty- two letters of their alphabet to represent 
the numbers 1, 2, . . 9, 10, 20, . . . 90, 100, ... 400 ; 
500 was 400 -(-100, etc. The thousands were repre 
sented by the symbols for units with a subscript 
comma at the right. J The Hebrew notation follows 
the same plan. 

The oldest Greek numerals (aside from the written 
words) were, in general, the initial letters of the funda- 
mental numbers. I for 1, n for 5 (ircwc), A for 10 
(ScKa), § and these were repeated as often as necessary. 

* Cantor, I., p. 84. t Cantor, I., p. 113. t Cantor. I., pp. 113-114. 

S Cantor, I., p. no. 


These numerals are described by the Byzantine gram- 
marian Herodianus (A. D. 200) and hence are spoken 
of as Herodianic numbers. Shortly after 500 B. C. 
two new systems appeared. One used the 24 letters 
of the Ionic alphabet in their natural order for the 
numbers from 1 to 24. The other arranged these 
letters apparently at random but actually in an order 
fixed arbitrarily; thus, a = l, )8 = 2, . . . . , » = 10, jc = 
20, . . . . , p = 100, o- = 200, etc. Here too there is 
no special symbol for the zero. 

The Roman numerals* were probably inheriteoB 
from the Etruscans. The noteworthy peculiarities 
are the lack of the zero, the subtractive principle 
whereby the value of a symbol was diminished by 
placing before it one of lower order (IV = 4, IX = 9J 
XL =40, XC==90), even in cases where the language 
itself did not signify such a subtraction ; and finally 
the multiplicative effect of a bar over the numerals 
(X]nf=30 000, c = 100 000). Also for certain frac- 
tions there were special symbols and names. Accord\ 
ing to Mommsen the Roman number-symbols I, V, i 
X represent the finger, the hand, and the doubl/ 
hand. Zangemeister proceeds from the standpoint 
that decern is related to decussare which means a 
perpendicular or oblique crossing, and argues that 
every straight or curved line drawn across the symbol 
of a number in the decimal system multiplies that 
number by ten. In fact, there are on monuments 

* Cantor, I., p. 486. 


representations of 1, 10, and 1000, as well as of 5 and 
500, to prove his assertion. * 

Of especial interest in elementary arithmetic is thd 
number-system of the Hindus, because it is to thesa 
Aryans that we undoubtedly owe the valuable position/ 
system now in use. Their oldest symbols for 1 to 9 
were merely abridged number-words, and the use of 
letters as figures is said to have been prevalent from 
the second century A. D.f The zero is of later origin ; 
its introduction is not proven with certainty till after 
400 A. D. The writing of numbers was carried on, 
chiefly according to the position-system, in various 
ways. One plan, which Aryabhatta records, repre- 
sented the numbers from 1 to 25 by the twenty-five 
consonants of the Sanskrit alphabet, and the succeed- 
ing tens (30, 40 ... . 100) by the semi-vowels and 
sibilants. A series of vowels and diphthongs formed 
multipliers consisting of powers of ten, ga meaning 
3, gi 300, gu 30 000, gau 3- 10". J In this there is no 
application of the position-system, although it ap- 
pears in two other methods of writing numbers in 
use among the arithmeticians of Southern India. 
Both of these plans are distinguished by the fact that 

*SxtMungsbtrichU der Berliner Akadetnie votn lo. November 1887. Words- 
worth, in his Fragments and Specimens of Early Latin, 1874, derives C for 
centum, M for mille, and L for quinguaginta from three letters of the Chal- 
cidian alphabet, corresponding to 0, ^, and x- He says: "The origin of this 
notation is, I believe, quite uncertain, or rather purely arbitrary, though, of 
course, we observe that the initials of tnille and centum determined the final 
shape taken by the signs, which at first were very different in form." 

tSee En^cUpadia Britannica, under " Numerals " 

^Cantor, I., p. 566. 


the same number can be made up in various ways. 
Rules of calculation were clothed in simple verse easy 
to hold in mind and to recall. For the Hindu mathe- 
maticians this was all the more important since they 
sought to avoid written calculation as far as possible. 
One method of representation consisted in allowing 
the alphabet, in groups of 9 symbols, to denote the 
numbers from 1 to 9 repeatedly, while certain vowels 
represented the zeros. If in the English alphabet ac- 
cording to this method we were to denote the num- 
bers from 1 to 9 by the consonants d, c, , . . z so that 
after two countings one finally has 5 = 2, and were to 
denote zero by every vowel or combination of vowels, 
the number 60502 might be indicated by siren or herotiy 
and might be introduced by some other words in the 
text. A second method employed type-words and 
combined them according to the law of position. 
Thus abdhi (one of the 4 seas) =4, surya (the sun 
with its 12 houses)=12, a^vin (the two sons of the 
sun)ss2. The combination abdhisurya^vinas denoted 
the number 2124.* 

Peculiar to the Sanskrit number-language are spe- 
cial words for the multiplication of very large num- 
bers. Arbuda signifies 100 millions, padma 10 000 
millions; from these are derived maharbuda^^X^^^ 
millions, fnahapadma^=\^^^^^ millions. Specially- 
formed words for large numbers run up to 10^^ and 
even further. This extraordinary extension of the 

* Cantor, I., p. sft;. 


decimal system in Sanskrit resembles a number-game, 
a mania to grasp the infinitely great. Of this endeavor 
to bring the infinite into the realm of number-percep- 
tion and representation, traces are found also among 
the Babylonians and Greeks. This appearance may 
find its explanation in mystic-religious conceptions or 
philosophic speculations. 

The ancient Chinese number-symbols are confineoV 
to a comparatively few fundamental elements arrangea 
in a perfectly developed decimal system. Here the 
combination takes place sometimes by multiplica- 
tion, sometimes by addition. Thus san=S, ^^^ = 10; 
chf san denotes 13, but san che 30^ Later, as a result 
of foreign influence, there arose two new kinds of no- 
tation whose figures show some resemblance to the 
ancient Chinese symbols. Numbers formed from 
^ them were not written from above downward but 
after the Hindu fashion from left to right beginning 
with the highest order. The one kind comprising the 
merchants' figures is never printed but is found only 
in writings of a busii^ess character. Ordinarily the 
ordinal and cardinal numbers are arranged in two 
lines one above another, with zeros when necessary, 
in the form of small circles. In this notation 

11=2, X = 4, j^ = 6, ^. = 10, /j=10 000, 0=0, 

» X 

and hence 7) O O -f-^_£^ = 20 046. 

♦Cantor, I., p. 630. 


Among the Arabs, those skilful transmitters of 
Oriental and Greek arithmetic to the nations of the 
West, the custom of writing out number-words con- 
tinued till the beginning of the eleventh century^ 
Yet at a comparatively early period they had already 
formed abbreviations of the number- words, the Divani 
figures. In the eighth century the Arabs -became ac- 
quainted ^\\\i the Hindu number-system and its fig- 
ures, including zero. From these figures there arose 
among the Western Arabs, who in their whole litera- 
ture presented a decided contrast to their Eastern re- 
latives, the Gubar 'numerals (dust-numerals) as vari- 
ants. These Gubar numerals, almost entirely forgotte^ 
to-day among the Arabs themselves, are the ancestors I 
of our modern numerals,* which are immediately dey 
rived from the apices of the early Middle Ages. These 
primitive ^yestern forms used in the abacus-calcula- 
tions are found in the West European MSS. of the 
eleventh and twelfth centuries and owe much of their 
prominence to Gerbert, afterwards Pope Sylvester II. 
(consecrated 999 A. D. ). 

The arithmetic of the Western nations, cultivated 
to a consid§,pable extent in the cloister- schools from 
the ninth century on, employed besides the abacus the 
Roman numerals, and consequently made no use of a 
symbol for zero. In Germany up to the year 1500 the 
Roman symbols were called German numerals in dis- 
tinction from the symbols — then seldom employed — 

* Hankel, p. 355. 


of Arab-Hindu origin, which included a zero (Arabic 
as-sifr, Sanskrit sunya, the void). The latter were 
called ciphers {Ziffern). From the fifteenth century on 
these Arab- Hindu numerals appear more frequently in 
Germany on monuments and in churches, but at that 
time they had not become common property.* The 
oldest monument with Arabic figures (in Katharein 
near Troppau) is said to date from 1007. Monuments 
of this kind are found in Pforzheim (1371), and in Ulm 
(1388). A frequent and free use of the zero in the 
thirteenth century is shown in tables for the calcula- 
tion of the tides at London and of the duration of 
moonlight, f In the year 1471 there appeared in Co- 
logne a work of Petrarch with page-numbers in Hindu 
figures at the top. In 1482 the first German arith 
metic with similar page-numbering was published in 
Bamberg. Besides the ordinary forms of numerals 
everywhere used to-day, which appeared exclusively 
in an arithmetic of 1489, the following forms for 4, 5, 
7 were used in Germany at the time of the struggle 
between the Roman and Hindu notations : 


The derivation of the modern numerals is illustrated 
by the examples below which are taken in succession 
from the Sanskrit, the apices, the Eastern Arab, the 

* Unger, Die Meihodik tUr prakiitcken Arithmttih^ 1888, p. 70. Hereafter 
referred to as Unger. 

tG^nlijer, Geschichte des mathemattschen Unterrichts im deutschen Mittel- 
alter bis zum Jahr 1525^ 1887, p. 175. Hereafter referred to as Giinther. 


Western Arab Gubar numerals, the numerals of the 
eleven th| thirteenth, and sixteenth centuries.* 

r !^" V A 

76 /\ 8 
ZS 7 % 

In the sixteenth century the Hindu position -arithX 
metic and its notation first found complete introduc^ 
tion among all the civilized peoples of the West. By 
this means was fulfilled one of the indispensable con- 
ditions for the development of common arithmetic in 
the schools and in the service of trade and commerce. 

* Cantor, table appended to Vol. 1, and Hankel, p. yi%. 



'T^HE simplest number-words and elementary count- 
^ ing have always been the common property of 
the people. Quite otherwise is it, however, with the 
different methods of calculation which are derived 
from simple counting, and with their application to 
complicated problems. As the centuries passed, that 
part of ordinary arithmetic which to-day every child 
knows, descended from the closed circle of particular 
castes or smaller communities to the common people, 
so as to form an important part of general culture. 
Among the ancients the education of the youth had to 
do almost wholly with bodily exercises. Only a riper 
age sought a higher cultivation through intercourse 
with priests and philosophers, and this consisted in 
part in the common knowledge of to-day: people 
learned to read, to write, to cipher. 

At the beginning of the first period in the historic 
development of common arithmetic stand the Egyp- 
tians. To them the Greek writers ascribe the inven- 
tion of surveying, of astronomy, and of arithmetic. To 
their literature belongs also the most ancient book on 


arithmetic, that of Ahmes, which teaches operations 
with whole numbers and fractions. The Babylonians 
employed a sexagesimal system in their position-arith- 
metic, which latter must also have served the pur- 
poses of a religious number-symbolism. The common 
arithmetic of the Greeks, particularly in most ancient 
times, was moderate in extent until by the activity of 
the scholars of philosophy there was developed a real 
mathematical science of predominantly geometric 
character. In spite of this, skill in calculation was 
not esteemed lightly. Of this we have evidence when 
Plato demands for his ideal state that the youth should 
be instructed in reading, writing, and arithmetic. 

The arithmetic of the Romans had a purely prac- 
tical turn ; to it belonged a mass of quite complicated 
problems arising from controversies regarding ques- 
tions of inheritance, of private property and of reim- 
bursement of interest. The Romans used duodecimal 
fractions. Concerning the most ancient arithmetic of 
the Hindus only conjectures can be made ; on the con- 
trary, the Hindu elementary arithmetic after the in- 
troduction of the position- system is known with toler- 
able accuracy from the works of native authors. The 
* Hindu mathematicians laid the foundations for the 
ordinary arithmetic processes of to-day. The influ- 
ence of their learning is perceptible in the Chinese 
arithmetic which likewise depends on the decimal sys- 
tem ; in still greater measure, however, among the 


Arabs who besides the Hindu numeral-reckoning also 
employed a calculation by columns. 

The time from the eighth to the beginning of the 
fifteenth century forms the second period. This is a 
noteworthy period of transition, an epoch of the trans- 
planting of old methods into new and fruitful soil, 
but also one of combat between the well-tried Hindu 
methods and the clumsy and detailed arithmetic ope- 
rations handed down from the Middle Ages. At 
first only in cloisters and cloister-schools could any 
arithmetic knowledge be found, and that derived from 
Roman sources. But finally there came new sugges- 
tions from the Arabs, so that from the eleventh to the 
thirteenth centuries there was opposed to the group of 
abacists, with their singular complementary methods, 
a school of algorists as partisans of the Hindu arith- 

Not until the fifteenth century, the period of in- 
vestigation of the original Greek writings, of the 
rapid development of astronomy, of the rise of the 
arts and of commercial relations, does the third pe- 
riod in the history of arithmetic begin. As early 
as the thirteenth century besides the cathedral and 
cloister-schools which provided for their own religious 
and ecclesiastical wants, there were, properly speak- 
ing, schools for arithmetic. Their foundation is to be 
ascribed to the needs of the brisk trade of German 
towns with Italian merchants who were likewise skilled 
computers. In the fifteenth and sixteenth centuries 


school affairs were essentially advanced by the human- 
istic tendency and by the reformation. Latin schools, 
writing schools, German schools (in Germany) for boys 
and even for girls, were established. In the Latin 
schools only the upper classes received instruction in 
arithmetic, in a weekly exercise : they studied the four 
fundamental rules, the theory of fractions, and at most 
the rule of three, which may not seem so very little 
when we consider that frequently in the universities 
of that time arithmetic was not carried much further. 
In the writing schools and German boys' schools the 
pupils learned something of calculation, numeration, 
and notation, especially the difference between the 
German numerals (in Roman writing) and the ciphers 
(after the Hindu fashion). In the girls' schools, which 
were intended only for the higher classes of people, no 
arithmetic was taught. Considerable attainments in 
computation could be secured only in the schools for 
arithmetic. The most celebrated of these institutions 
was located at Nuremberg. In the commercial towns 
there were accountants' guilds which provided for the 
extension of arithmetic knowledge. But real mathe- 
maticians and astronomers also labored together in de- 
veloping the methods of arithmetic. In spite of this 
assistance from men of prominence, no theory of arith- 
metic instruction had been established even as late as 
in the sixteenth century. What had been done be- 
fore had to be copied. In the books on arithmetic 


were found only rules and examples, almost never 
proofs or deductions. 

The seventeenth century brought no essential 
change in these conditions. Schools existed as before 
where they had not been swallowed up by the horrors 
of the Thirty- Years' War. The arithmeticians wrote 
their books on arithmetic, perhaps contrived calculat- 
ing machines to make the work easier for their pupils, 
or composed arithmetic conversations and poems. A 
specimen of this is given in the following extracts 
from Tobias Beutel's Arithmetical the seventh edition 
of which appeared in 1693.* 

" Numerieren lehrt im Rechen 
Zahlen schreiben und aussprechen.** 

"In Summen bringen heisst addieren 
Dies muss das WOrtlein Und vollf tihren. ** 

•• Wie eine Hand an uns die andre wglschet rein 
Kann eine Species der andem Probe seyn.'* 

* • We are taught in numeration 
Number writing and expression,** 
etc., etc. 

Commercial arithmetic was improved by the cultiva 
tion of the study of exchange and discount, and the 
abbreviated method of multiplication. The form of 
instruction remained the same, i. e., the pupil reck- 
oned according to rules without any attempt being 
made to explain their nature. 

The eighteenth century brought as its first and 

* UnRer, p. 124. 


most important innovation the statutory regulation of 
school matters by special school laws, and the estab- 
lishment of normal schools (the first in 1732 at Stet- 
tin in connection with the orphan asylum). As reor- 
ganizers of the higher schools appeared the pietists 
and philanthropinists. The former established Real- 
schulen (the oldest founded 1738 in Halle) and higher 
Biirgerschulen; the latter in their 5M«/^« der Aufklarung 
sought by an improvement of methods to educate 
cultured men of the world. The arithmetic exercise- 
books of this period contain a simplification of divi- 
sion (the downwards or under-itself division) as well 
as a more fruitful application of the chain rule and 
decimal fractions. By their side also appear manuals 
of method whose number is rapidly increasing in the 
nineteenth century. In these, elementary teaching 
receives especial attention. According to Pestalozzi 
(1803) the foundation of calculation is sense- percep- 
tion, according to Grube (1842), the comprehensive 
treatment of each number before taking up the next, 
according to Tanck and Knilling (1884), counting. 
In Pestalozzi's method "the decimal structure of our 
number-system, which includes so many advantages 
in the way of calculation, is not touched upon at all, 
addition, subtraction, and division do not appear as 
separate processes, the accompanying explanations 
smother the principal matter in the propositions, that 
is the arithmetic truth."* Grube has simply drawn 

* Unger, p. 179. 



from Pestalozzi's principles the most extreme oondu- 
sions. His sequence <<is in many respects faulty; his 
processes unsuitable. " * The historical development 
of arithmetic speaks in favor of the counting-prin- 
ciple : the first reckoning in every age has been an 
observing and counting. 




/• The Arithmetic of Whole Numbers. 

If we leave out of account finger-reckoning, which 
cannot be shown with absolute certainty, then accord- 
ing to a statement of Herodotus the ancient Egyptian 
computation consisted of an operating with pebbles on 
a reckoning-board whose lines were at right angles to 
the computer. Possibly the Babylonians also used a 
similar device. In the ordinary arithmetic of the latter, 
as among the Egyptians, the decimal system prevails, 
but by its side we also find, especially in dealing with 
fractions, a sexagesimal system. This arose without 
doubt in the working out of the astronomical observa- 
tions of the Babylonian priests, f The length of the 
year 6f 360 days furnished the occasion for the divi- 
sion of the circle into 360 equal parts, one of which 
was to represent the apparent daily path of the sun 
upon the celestial sphere. If in addition the construc- 

* Uoger. pp. 192, igs. t Cantor, I., p. 8a 


tion of the regular hexagon was known, then it was 
natural to take every 60 of these parts again as units. 
The number 60 was called soss. Numbers of the 
sexagesimal system were again multiplied in accord- 
ance with the rules of the decimal system : thus a fur 
= 600, a xar=3600. The sexagesimal system estab- 
lished by the Babylonian priests also entered into 
their religious speculations, where each of their divin- 
ities was designated by one of the numbers from 1 to 
60 corresponding to his rank. Perhaps the Babyloni- 
ans also divided their days into 60 equal parts as has 
been shown for the Veda calendars of the ancient 

The Greek elementary mathematics, at any rate 
as early as the time of Aristophanes (420 B. C.),* used 
finger-reckoning and reckoning-boards for ordinary 
computation. An explanation of the finger-reckoning 
is given by Nicholas Rhabdaf of Smjrrna (in the four- 
teenth century). Moving from the little finger of the 
left hand to the little finger of the right, three fingers 
were used to represent units, the next two, tens, the 
next two, hundreds, and the last three, thousands. 
On the reckoning board, the abax {^pai, dust board), 
whose columns were at right angles to the user, the 
operations were carried on with pebbles which had a 
different place-value in different lines. Multiplication 
was performed by beginning with the highest order in 
each factor and forming the sum of the partial pro- 

♦ Castor, I., pp. I«^ 479* t Gow. History cf Greek MathemaHet, p. 14. 



ducts. Thus the calculation was effected (in modern 
form) as follows: 

126 -237 = (100 + 20 + 6) (200 + 30 + 7) 
= 20 000+ 3000 +700 

+ 4 000+ 600 +140 

+ 1200+ 180 + 42 

= 29 862 

According to Pliny, the finger-reckoning of the 
Romans goes back to King Numa ; * the latter had 
made a statue of Janus whose fingers represented the 
number of the days of a year (355). Consistently with 
this Boethius calls the numbers from 1 to 9 finger- 
numbers, 10, 20, 30, . . . joint-numbers, 11, 12, . . . 
19, 21, 22, . . . 29, . . . composite numbers. In ele 

^7 d$ b^ bi 


t I I t t i t t i 

ixl ® <ai) <D c X I e t 










mentary teaching the Romans used the abaais, a 
board usually covered with dust on iVhich one could 

* Cantor, I., P- 491. 


trace figures, draw columns^ and work with pebbles. 
Or if the abacus was to be used for computing only, 
it was made of metal and provided with grooves (the 
vertical lines in the schematic drawing on the pre- 
ceding page) in which arbitrary marks (the cross- 
lines) could be shifted. 

The columns ai . . . a^, b\ , , . ^ form a system 
from 1 to 1 000 000 ; upon a column a are found four 
marks, upon a column b only one mark. Each of the 
four marks represents a unit, but the upper single 
mark five units of the order under consideration. 
Further a mark upon ri=^, upon ^t=r^, upon k% 

= A» ^po^ ^4 = A' ^PO'^ ^» = tjV (i^clative to the di- 
vision of the tf's). The abacus of the figure represents 
the number 782 192 + ^ + ^ + ^ = 782 192||. This 
ahacus served for the reckoning of results of simple 
problems. Along with this the multiplication-table 
was also employed. For larger multiplications there 
were special tables. Such a one is mentioned by Vic- 
torius (about 450 A. D.).* From Boethius, who calls 
the abacus marks apices, we learn something about 
multiplication and division. Of these operations the 
former probably, the latter certainly, was performed 
by the use of complements. In Boethius the term 
differentia is applied to the complement of the divisor 
to the next complete ten or hundred. Thus for the 
divisors 7, 84, 213 the differentiae are 3, 6, 87 f respec- 
tively. The essential characteristics of this comple- 

* Cantor, I., p. 493. t Cantor, I , i». S44. 


mentary division are seen from the following example 
put in modem form : 

2»^ 257 ^^^ e^±V! 117 

14 20 — 6 ' 20 — 6 ' 20 — 6 
117 ^ . 30 + 17 ^ . 47 

20—6 ' 20 — 6 ' 20—6 

20—6 ' 20—6 ' 20—6 

1? = 1+ A 

14 ^ 14 

The swanpan of the Chinese somewhat resembles 
the abacus of the Romans. This calculating machine 
consists of a frame ordinarily with ten wires inserted. 
A cross wire separates each of the ten wires into two 
unequal parts ; on each smaller part two and on each 
larger five balls are strung. The Chinese arithmetics 
give no rules for addition and subtraction, but do for 
multiplication, which, as with the Greeks, begins 
with the highest order, and fordivision, which appears 
in the form of a repeated subtraction. 

The calculation of the Hindus, after the introduc- 
tion of the arithmetic of position, possessed a series 
of suitable rules for performing the fundamental ope- 
rations. In the case of a smaller figure in the minu- 
end subtraction is performed by borrowing and by 
addition (as in the so-called Austrian subtraction).'*' 

*The Auitrian subtraction corresponds in put to the ataal method of 
<* making change." 



In multiplication, for which several processes are 
available, the product is obtained in some cases 
by separating the multipliers into factors and subse- 
quently adding the partial products. In other cases 
a schematic process is introduced whose peculiarities 
are shown in the example 315 '37 = 11 655, 




3 / 



The result of the multiplication is obtained by the 
addition of the figures found within the rectangle in 
the direction of the oblique lines. With regard to 
division we have only a few notices. Probably, how- 
ever, complementary methods were not used. 

The earliest writer giving us information on the 
arithmetic of the Arabs is Al Khowarazmi. The bor- 
rowing from Hindu arithmetic stands out very clearly. 
Six operations were taught. Addition and subtraction 
begin with the units of highest order, therefore on 
the left ; halving begins on the right, doubling again 
on the left. Multiplication is effected by the process 
which the Hindus called Tatstha (it remains stand- 
ing).* The partial products, beginning with the high- 
est order in the multiplicand, are written above the 
corresponding figures of the latter and each figure 

« Cantor, L. p. 674, n\. 


of the product to which other units from a later par- 
tial product are added (in sand or dust), rubbed out 
and corrected, so that at the end of the computation 
the result stands above the multiplicand. In divi- 
sion, which is never performed in the complementary 
fashion, the divisor stands below the dividend and 
advances toward the right as the calculation goes on. 
Quotient and remainder appear abova the divisor in 
4^:=28^J, somewhat as follows:* 



Al Nasawif also computes after the same fashion as 
Al Khowarazmi. Their methods characterise the ele- 
mentary arithmetic of the Eastern Arabs. 

In essentially the same manner, but with more or 
less deviation in the actual work, the Western Arabs 
computed. Besides the Hindu figure-computation 
Ibn al Banna teaches a sort of reckoning by columns. | 
Proceeding from right to left, the columns are com- 
bined in groups of three ; such a group is called ta- 
karrur; the number of all the columns necessary to 
record a number is the mukarrar. Thus for the num- 
ber 3 849 922 the takarrur or number of complete 
groups is 2, the mukarrar r=z1 , Al Kalsadi wrote a 

♦Cantor, I., p. 674. t Cantor, I., p. 716. t Cantor, I., p. 757. 


work Raising of the Veil of the Science of Gubar,* The 
original meaning of Gubar (dust) has here passed 
over into that of the written calculation with figures. 
Especially characteristic is it that in addition, sub- 
traction (=ztarh^ taraha =to throw away) and multi- 
plication the results are written above the numbers 
operated upon, as in the following examples : 

193 + 45=238 and 238 — 193 = 46 

is written, is written, 

238 J^ 

193' 238* 

45 193 

Several rules for multiplication are found in Al Kal- 
sadi, among them one with an advancing multiplier. 
In division the result stands below. 


7-143 = 1001 1001 

is written, 1001 7 

= 143 

21 is written, 32 
28 1001 

7 777 

143 143 


2. Calculation With Fractions, 

In his arithmetic Ahmes gives a large number of 
examples which show how the Egyptians dealt with 
fractions. They made exclusive use of unit-fractions, 

* Cantor, I., p. 763. 



i. e.y fractions with numerator 1. For this numerator, 
therefore, a special symbol is found, in the hiero- 
glyphic writing o, in the hieratic a point, so that in 
the latter a unit fraction is represented by its denomi- 
nator with a point placed above it. Besides these 
there are found for ^ and ^ the hieroglyphs i and 

jf ; * in the hieratic writing there are likewise special 
symbols corresponding to the fractions ^, f , ^, and ^. 
The first problem which Ahmes solves is this, to sep- 
arate a fraction into unit fractions. £. g., he finds 

* = i + A> A = iAF + Tiv + ^- This separation, 
really an indeterminate problem, is not solved by 
Ahmes in general form, but only for special cases. 

The fractions of the Babylonians being entirely 
in the sexagesimal system, had at the outset a com- 
mon denominator, and could be dealt with like whole 
numbers. In the written form only the numerator 
was given with a special sign attached. The Greeks 
wrote a fraction so that the numerator came first with 
a single stroke at the right and above, followed in the 
same line by the denominator with two strokes, writ- 
ten twice, thus t^'ica'ica" = J^. In unit fractions the 
numerator was omitted and the denominator written 
only once: 8" = J. The unit fractions to be added 
follow immediately one after another. f f kjj" pi^' a-Kh" 

= i + ^ + Ths + ish = itih' I^ arithmetic proper, 
extensive use was made of unit- fractions, later also of 

*For carefnlly drawn symbols see Cantor, I. p. 45. 
t Cantor, I., p. 1x8. 


sexagesimal fractions (in the computiation of angles). 
Of the use of a bar between the terms of a fraction 
there is nowhere any mention. Indeed, where such 
use appears to occur, it marks only the result of an 
addition, but not a division.* 

The fractional calculations of the Romans furnish 
an example of the use of the duodecimal system. 
The fractions {minutict) ^, A» • • * -H ^^^ special 
names and symbols. The exclusive use of these duo- 
decimal fractions t was due to the fact that the as, 
a mass of copper weighing one pound, was divided 
into twelve unciiB. The uncia had four skilici and 
twenty- four .f^r/^«/r. l=aj, \-=i semis, ^=ztritns, J = 
quadrans, etc. Besides the twelfths special names 
were given to the fractions ^, -^^ i^, y^^, ^^ The 
addition and subtraction of such fractions was com- 
paratively simple, but their multiplication very de- 
tailed. The greatest disadvantage of this system con- 
sisted in the fact that all divisions which did not fit 
into this duodecimal system could be represented by 
miautise either with extreme difficulty or only approxi- 

In the computations of the Hindus both unit frac- 
tions and derived fractions likewise appear. The de- 
nominator stands under the numerator but is not sep- 
arated from it by a bar. The Hindu astronomers 
preferred to calculate with sexagesimal fractions. In 
the computations of the Arabs Al Khowarazmi gives 

* Tannery in Bibl. Math. t Hankel, p. 57. Doubted by Cantor, I. p. 49a 


special words for half, third, . . . ninth (expressible 
fractions).* All fractions with denominators non-divis- 
ible by 2, 3, . . . 9, are called mute fractions ; they 
were expressed by a circumlocution, e. g., -^ as 2 
parts of 17 parts. Al Nasawi writes mixed numbers 
in three lines, one under another, at the top the whole 
number, below this the numerator, below this the de- 
nominator. For astronomical calculations fractions 
of the sexagesimal system were used exclusively. 

J. Applied Arithmetic, 

The practical arithmetic of the ancients included 
besides the common cases of daily life, astronomical 
and geometrical problems. The latter will be passed 
over here because they are mentioned elsewhere. In 
Ahmes problems in partnership are developed and 
also the sums of some of the simplest series deter- 
mined. Theon of Alexandria showed how to obtain 
approximately the square root of a number of angle 
degrees by the use of sexagesimal fractions and the 
gnomon. The Romans were concerned principally 
with problems of interest and inheritance. The Hin- 
dus had already developed the method of false posi- 
tion {Regula falsi) and the rule of three, and made 
a study of problems of alligation, cistern-filling, and 
series, which were still further developed by the Arabs. 
Along with the practical arithmetic appear frequent 

♦Cantor. I., p. 675. 


traces of observations on the theory of numbers. The 
Egyptians knew the test of divisibility of a number by 
2. The Pythagoreans distinguished numbers as odd 
and even, amicable, perfect, redundant and defective.* 
Of two amicable numbers each was equal to the sum 
of the aliquot parts of the other (220 gives 1 + 2 -f 4 
+ 5 + 10+11 + 20 + 22 + 44+55 + 110 = 284 and 
284 gives 1 + 2 + 4 + 71 + 142 = 220). A perfect num- 
ber was equal to the sum of its aliquot parts (6 = 1 + 
2 + 3). If the sum of the aliquot parts was greater or 
less than the number itself, then the latter was called 
redundant or defective respectively (8 > 1 + 2 + 4 ; 12 
<l + 2 + 3 + 4+6). Besides this, Euclid starting 
from his geometric standpoint commenced some fun- 
damental investigations on divisibility, the* greatest 
common measure and the least common multiple. 
The Hindus were familiar with casting out the nines 
and with continued fractions, and from them this 
knowledge went over to the Arabs. However insig- 
nificant may be these beginnings in their ancient 
form, they Contain the germ of that vast development 
in the theory of numbers which the nineteenth cen- 
tury has brought about. 

'■'Cantor, 1.. p. 156. 




/. The Arithmetic of Whole Numbers. 

In the cloister schools, the episcopal schools, and 
the private schools of the Merovingian and Carloving- 
ian period it was the monks almost exclusively who 
gave instruction. The cloister schools proper were of 
only slight importance in the advancement of mathe- 
matical knowledge : on the contrary, the episcopal 
and private schools, the latter based on Italian meth- 
ods, seem to have brought very beneficial results. 
The first to foreshadow something of the mathemat- 
* ical knowledge of the monks is Isidorus of Seville. 
This cloister scholar confined himself to making con-, 
jectures regarding the derivation of the Roman nu- 
merals, and says nothing at all about the method of 
computation of his contemporaries. The Venerable 
Bede likewise published only some extended observa- 
tions on finger-reckoning. He shows how to repre- 
sent numbers by the aid of the fingers, proceeding 
from left to right, and thereby assumes a certain ac- 
quaintance with finger- reckoning, mentioning as his 
predecessors Macrobius and Isidorus. * This calculus 
digitalis^ appearing in both the East and the West in 

* Cantor, I., p. 776. 


exactly the same fashion, played an important part in 
fixing the dates of church feasts by the priests of that 
time ; at least computus digitalis and computus ecclesias- 
ticus were frequently used in the same sense.* 

With regard to the fundamental operations proper 
Bade does not express himself. Alcuin makes much 
of number- mysticism and reckons in a very cumbrous 
manner with the Roman numerals, t Gerbert was the 
first to give in his Regula de abaco computi actual rules, 
in which he depended upon the arithmetic part of 
Boethius*s work. What he teaches is a pure abacus- 
reckoning, which was widely spread by reason of his 
reputation. Gerbert's abacus, of which we have an 
accurate description by his pupil Bernelinus, was a 
table which for the drawing of geometric figures was 
sprinkled with blue sand, but for calculation was di- 
vided into thirty columns of which three were reserved 
for fractional computations. The remaining twenty- 
seven columns were separated from right to left into 
groups of three. At the head of each group stood like- 
wise from right to left S [singularis^^ D {decern^, C {cen- 
turn). The number-symbols used, the so-called apices, 
are symbols for 1 to 9, but without zero. In calcu- 
lating with this abacus the intermediate operations 
could be rubbed out, so that finally only the result re- 
mained; or the operation was made with counters. 
The fundamental operations were performed princi- 
pally by the use of complements, and in this respect 

♦ Gflnther. t Gunther. 




division is especially characteristic. The formation 
of the quotient i|A = 33^ will explain this comple- 
mentary division. 





































» 1 






• 1 


19 9 









. 13 






In the example given the complete performance of the com- 
plementary division stands on the left ; the figures to be rubbed 
out as the calculation goes on are indicated by a period on the 
right. On the right is found the abacus-division without the for- 
mation of the difiFerence in the divisor, below it the explanation of 
the complementary division in modem notation. 


In the tenth and eleventh centuries there appeared 
a large number of authors belonging chiefly to the 
clergy who wrote on abacus-reckoning with apices 
but without the zero and without the Hindu-Arab 
methods. In the latter the apices were connected with 
the abacus itself or with the representation of num- 
bers of one figure, while in the running text the Roman 
numeral symbols stood for numbers of several figures. 
The contrast between the apices- plan and the Roman 
is so striking that Oddo, for example, writes : "If one 
takes 5 times 7, or 7 times 5, he gets XXXV '' (the 5 
and 7 written in apices).* 

At the time of the abacus-reckoning there arose the peculiar 
custom of representing by special signs certain numbers which do 
not appear in the Roman system of symbols, and this use contin- 
ued far into the Middle Ages. Thus, for example, in the town- 
books of Greifswald 250 is continually represented by CCC^ ' 

The abacists with their remarkable methods of di- 
vision completely dominated Western reckoning up 
to the beginning of the twelfth century. But then a 
complete revolution was effected. The abacus, the 
heir fif the computus^ i. e., the old Roman method of 
calculation and number-writing, was destined to give 
way to the algorism with its sensible use of zero and 
its simpler processes of reckoning, but not without a 
further struggle. J People became pupils of the Wes- 
tern Arabs. Among the names of those who extended 

*Cantor, I., p.846. t Giinther p. 175. $ Giinther, p. 107. 



Arab methods of calculation stands forth especially 
pre-eminent that of Gerhard of Cremona, because he 
translated into Latin a series of writings of Greek 
and Arab authors.* Then was formed the school of 
algorists who in contrast to the abacists possessed no 
complementary division but did possess the Hindu 
place -system with zero. The most lasting material 
for the extension of Hindu methods was furnished by 
Fibonacci in his Liber abaci. This book "has been 
the mine from which arithmeticians and algebraists 
have drawn their wisdom ; on this account it has be- 
come in general the foundation of modern science, "t 
Among other things it contains the four rules for 
whole numbers and fractions in detailed form. It is 
worthy of especial notice that besides ordinary sub- 
traction with borrowing he teaches subtraction by in- 
creasing the next figure of the subtrahend by one, 
and that therefore Fibonacci is to be regarded as the 
creator of this elegant method. 

2, Arithmetic of Fractions, 

Here, also, after Roman duodecimal fractions had 
been exclusively cultivated by the abacists Beda, Ger- 
bert and Bernelinus, Fibonacci laid a new foundation 
in his exercises preliminary to division. He showed 
how to separate a fraction into unit fractions. Espe- 
cially advantageous in dealing with small numbers 

* Hankel, p. 336. t Hankel, p. 343. 


is his method of determining the common denomina- 
tor: the greatest denominator is multiplied by each 
following denominator and the greatest common meas- 
ure of each pair of factors rejected. (Example : the 
least common multiple of 24, 18, 15, 9, 8, 5 is 24*3*5 
= 360.) 

J. Applied Arithmetic, 

The arithmetic of the abacists had for its main 
purpose the determination of the date of Easter. Be- 
sides this are found, apparently written by Alcuin, 
Problems for Quickening the Mind which suggest Ro- 
man models. In this department also Leonardo Fibo- 
nacci furnishes the most prominent rule (the regula 
falsi), but his problems belong more to the domain of 
algebra than to that of lower arithmetic. 

Investigations in the theory of numbers could 

hardly be expected from the school of abacists. On 

the other hand, the algorist Leonardo was familiar 

with casting out the nines, for which he furnished an 

ndependent proof. 



7. The Arithmetic of Whole Numbers, 

While on the whole the fourteenth century had 
only reproductions to show, a new period of brisk ac- 


tivity begins with the fifteenth century, marked by 
Peurbach and Regiomontanus in Germany, and by 
Luca Pacioli in Italy. As far as the individual pro- 
cesses are concerned, in addition the sum sometimes 
stands above the addends, sometimes below; subtrac- 
tion recognizes "carrying" and ** borrowing '* ; in 
multiplication various methods prevail ; in division no 
settled method is yet developed. The algorism of 
Peurbach names the following arithmetic operations : 
Numeratio, additio^ subtraction mediation duplatiOy multi- 
plication divisio^ progressio (arithmetic and geometric 
series), besides the extraction of roots which before the 
invention of decimal fractions was performed by the 
aid of sexagesimal fractions. His upwards- division 
still used the arrangement of the advancing divisor ; 
it was performed in the manner following (on the left 
the explanation of the process, on the right Peurbach's 
division, where figures to be erased in the course of 
the reckoning are indicated by a period to the right 
and below) : The oral statement would be somewhat 
like this: 36 in 84 twice, 2-3 = 6, 8— 6 = 2, written 
above 8; 2-6 = 12, 24—12 = 12, write above, strike 
out 2, etc. The proof of the accuracy of the result is 
obtained as in the other operations by casting out the 
nines. This method of upwards-division which is not 
difiicult in oral presentation is still found in arith- 
metics which appeared shortly before the beginning 
of the nineteenth century. 

ARITUM£'iIC» 43 




g I.O.4. 

^^ 8 4 7 9 I 235 




8 4 7 9 
8 6 6 6 



In the sixteenth century work in arithmetic had 
entered the Latin schools to a considerable extent ; but 
to the great mass of children of the common people 
neither school men nor statesmen gave any thought 
before 1525. The first regulation of any value in this 
line is the Bavarian Schuelordnungk de anno 1^48 which 
introduced arithmetic as a required study into the vil- 
lage schools. Aside from an occasional use of finger- 
reckoning, this computation was either a computation 
upon lines with counters or a figure-computation. In 
both cases the work began with practice in numeration 
in figures. To perform an operation with counters a 
series of horizontal parallels was drawn upon a suit- 
able base. Reckoned from below upward each counter 
upon the 1st, 2d, 3d, . . . line represented the value 
1, 10, 100, . . ., but between the lines they represented 
5, 50, 500, . . . The following figure shows the rep- 


resentation of 41 096^. In subtraction the minuend, 
in multiplication the multiplicand was put upon the 
lines. Division was treated as repeated subtractions. 
This line- reckoning was completely lost in the seven- 

O O O O 

^4— & 

oSo o o 



teenth century when it gave place to real written 
arithmetic or figure-reckoning by which it had been 
accompanied in the better schools almost from the 


In the ordinary business and trade of the Middle 
Ages use was also made of the widely-extended score- 
reckoning. At the beginning of the fifteenth century 
this method was quite usual in Frankfort on the Main, 
and in England it held its own even into the nine- 
teenth century. Whenever goods were bought of a 
merchant on credit the amount was represented by 
notches cut upon a stick which was split in two length- 
wise so that of the two parts which matched, the debtor 
kept one and the creditor one so that both were se- 
cured against fraud.* 

In the cipher-reckoning the computers of the six- 
teenth century generally distinguished more than 4 
operations; some counted 9, i. e., the 8 named by 

* Cantor, M. Matkem. Bgitr. »um KuUurUben d*r V9Uur, Halle, 1863. 


Peurbach and besides, as a ninth operation, evolution, 
the extraction of the square root by the formula (« +^)* 
=tf* + 2tf^ + ^, and the extraction of the cube root 
by the formula (a + ^)» = a» + (a + b) Zab + ^•. Defi- 
nitions appeared, but these were only repeated circum- 
locutions. Thus Gram mateus says : '' Multiplication 
shows how to multiply one number by the other. 
Subtraction explains how to subtract one number 
from the other so that the remainder shall be seen.*'* 
Addition was performed just as is done to-day. In 
subtraction for the case of a larger figure in the sub- 
trahend, it was the custom in Germany to complete 
this figure to 10, to add this complement to the min- 
uend figure, but at the same time to increase the figure 
of next higher order in the subtrahend by 1 (Fibo- 
nacci's counting-on method). In more comprehen- 
sive books, borrowing for this case was also taught. 
Multiplication, which presupposed practice in the mul- 
tiplication table, was performed in a variety of ways. 
Most frequently it was effected as to-day with a des- 
cent in steps by movement toward the left. Luca 
Pacioli describes eight different kinds of multiplica- 
tion, among them those above mentioned, with two 
old Hindu methods, one represented on p. 29, the 
other cross-multiplication or the lightning method. 
In the latter method there were grouped all the pro- 
ducts involving units, all those involving tens, all 
those involving hundreds. The multiplication 

♦ Uager, p. 7a. 



243-139=9-3 + 10(9-4 + 3-3)+ 100(9-2 + 3-4 + 1-3) 
+ 1000(2-3 + l-4)+10 000-2-l 

was represented as follows : 

2 4 8 

In German books are found, besides these, two note- 
worthy methods of multiplication, of which one be- 
gins on the left (as with the Greeks), the partial pro- 
ducts being written in succession in the proper place, 
as shown by the following example 243*839: 





839-243 = 2-8-10* + 2-3-10« + 2-9-102 
4- 4-8-108 + 4-3-102 + 4-9 -10 
+ 3-810« + 3-3-10 +3-9. 

In division the upwards- division prevailed ; it was 
used extensively, although Luca Pacioli in 1494 taught 
the downwards-division in modern form. 

After the completion of the computation, in con- 
formity to historical tradition, a proof was demanded. 
At first this was secured by casting out the nines. 
On account of the untrustworthiness of this method, 
which Pacioli perfectly realised, the performance of 


the inverse operation was recommended. In course 
of time the use of a proof was entirely given up. 

Signs of operation properly so called were not 
yet in use; in the eighteenth century they passed 
from algebra into elementary arithmetic. Widmann, 
however, in his arithmetic has the signs + and — , 
which had probably been in use some time amdng the 
merchants, since they appear also in a Vienna MS. of 
the fifteenth century.* At a later time Wolf has the 
sign -i- for minus. In numeration the first use of the 
word ** million" in print is due to Pacioli {Summa de 
Arithmetical i494)- Among the Italians the word "mil- 
lion " is said originally to have represented a concrete 
mass, viz., ten tons of gold. Strangely enough, the 
words "byllion, tryllion, quadrillion, quyllion, sixlion, 
septyllion, ottyllion, nonyllion," as well as ** million,*' 
are found as early as 1484 in Chuquet, while the word 
"miliars" (equal to 1000 millions) is to be traced 
back to Jean Trenchant of Lyons (1588). f 

The seventeenth century was especially inventive 
in instrumental appliances for the mechanical per- 
formance of the fundamental processes of arithmetic. 
Napier's rods sought to make the learning of the mul- 
tiplication-table superfluous. These rods were quad- 
rangular prisms which bore on each side the small 
multiplication-table for one of the numbers 1, 2, ... 9. 

*Gerhardt, Geschichtg der Mathemattk in Deutschland^ 1877. Hereafter 
referred to as Gerhardt. Statement now shown to be incorrect. 

tMUUer. Historisck-etyniologiMche Studien Uirr mathetnattsckt TerminO' 
logie. Hereafter referred to as Miiller 


For extracting square and cube roots rods were used 
with the squares and cubes of one-figure numbers in- 
scribed upon them. Real calculating machines which 
gave results by the simple turning of a handle, but on 
that account must have proved elaborate and expen- 
sive, were devised by Pascal, Leibnitz, and Matthftus 
Hahn (1778). 

A simplification of another kind was effected by 
calculating-tables. These were tables for solving 
problems, accompanied also by very extended multi- 
plication-tables, such as those of Herwart von Hohen- 
burg, from which the product of any two numbers 
from 1 to 999 could be read immediately. 

For the methods of computation of the eighteenth 
century the arithmetic writings of the two Sturms, 
and of Wolf and Kastner, are of importance. In the 
interest of commercial arithmetic the endeavor was 
made to abbreviate multiplication and division by 
various expedients. Nothing essentially new was 
gained, however, unless it be the so-called mental 
arithmetic or oral reckoning which in the later decades 
of this period appears as an independent branch. 

The nineteenth century has brought as a novelty 
in elementary arithmetic only the introduction of the 
so-called Austrian subtraction (by counting on) and 
division, methods for which Fibonacci had paved the 
way. The difference 323 — 187 = 136 is computed 
by saying, 7 and 6, 9 and 3, 2 and 1 ; and 43083 : 185 
is arranged as in the first of the following examples : 






















With sufficient practice this process certainly secured 

a considerable saving of time, especially in the case 

of the determination of the greatest common divisor 

of two or more numbers as shown by the second of 

the above examples 

1679 28 73 

2737 ~ll9' 

J. Arithmetic of Fractions. 

At the beginning of this period reckoning with 
fractions was regarded as very difficult. The pupil 
was first taught how to read fractioms: ''It is to be 
noticed that every fraction has two figures with a line 
between. The upper is called the numerator, the 
lower the denominator. The expression of fractions is 
then: name first the upper figure, then the lower, with 
the little word part as \ part" (Grammateus, 1518).* 
Then came rules for the reduction of fractions to a 
common denominator, for reduction to lowest terms, 
for multiplication and division ; in the last the fractions 
were first made to have a common denominator. Still 
more is found in Tartaglia who knew how to find the 
least common denominator ; in Stifel who performed 

* Uoger, p. 84. 


division by a fraction by the use of its reciprocal, and 
in the works of other writers. 

The way for the introduction of decimal fractions 
was prepared by the systems of sexagesimal and duo- 
decimal fractions, since by their employment opera- 
tions with fractions can readily be performed by the 
corresponding operations with whole numbers. A no- 
tation such as has become usual in decimal fractions 
was already known to Rudolff,* who, in the division 
of integers by powers of 10, cuts off the requisite 
number of places with a comma. The complete knowl- 
edge of decimal fractions originated with Simon Stevin 
who extended the position-system below unity to any 
extent desired. Tenths, hundredths, thousandths, . . . 
were called primes, sekondes, terzes . . . ; 4. 628 is writ- 
ten 4(0) 6(1) 2(2) 8(8). Joost Burgi, in his tables of sines, 
perhaps independently of Stevin, used decimal frac- 
tions in the form 0.32 and 3.2. The introduction of 
the comma as a decimal point is to be assigned to 
Kepler. "j* In practical arithmetic, aside from logarith- 
mic computations, decimal fractions were used only 
in computing interest and in reduction-tables. They 
were brought into ordinary arithmetic at the begin- 
ning of the nineteenth centur}' in connection with the 
introduction of systems of decimal standards. 

*Gerhardt. "» "J 

tThe first use of the decimal point is found in the trigonometric tables 
of Pitiscus, i6i2. Cantor, II., p. 555. 


J. Applied Arithmetic. 

During the transition period of the Middle Ages 
applied arithmetic had absorbed much from the Latin 
treatises in a superficial and incomplete manner ; the 
fifteenth and sixteenth centuries show evidences of 
progress in this direction also. Even the Bamberger 
Arithmetic of 1483 bears an exclusively practical stamp 
and aims only at facility of computation in mercan- 
tile affairs. That method of solution which in the 
books on arithmetic everywhere occupied the first 
place was the **regeldetri"-(r<^«/a de tri, rule of 
three), known also as the ** merchant's rule,'' or 
''golden rule."* The statement of the rule of three 
was purely mechanical ; so little thought was bestowed 
upon the accompanying proportion that even master 
accountants were content to write 4 fl 12 lb 20 fl? in- 
stead of 4 fl : 20 fl = 12 ft) : ^ Ib.f There can indeed 
be found examples of the rule of three with indirect 
ratios, but with no explanations of any kind whatever. 
Problems involving the compound rule of three {regula 
de quinque^ etc. ) were solved merely by successive ap- 
plications of the simple rule of three. In Tartaglia 
and Widmann we find equation of payments treated 
according to the method still in use to-day. Other- 
wise, Widmann 's Arithmetic of 1489 shows great ob- 
scurity and lack of scope in rules and nomenclature, 
so that not infrequently the same matter appears un- 

♦ Cantor, II., p. 205 : Unger, p. 86. t Cantor, 1 1., p. 368; Unger, p. 87. 


der different names. He introduces ''Regula Residui, 
Reciprocationis, Excessus, Divisionis^ Quadrata, In- 
ventionis, Fusti, Transversa, Ligar, Equalitatis, Legis, 
Augment!, Augmenti et Decrement!, Sententiarum, 
Suppositionis, Collectionis, Cubica, Lucri, Pagamenti, 
Alligationis, Falsi/' so that in later years Stifel did not 
hesitate to declare these things simply laughable.* 
Problems of proportional parts and alligation were 
solved by the use of as many proportions as corre- 
sponded to the number of groups to be separated. 
For the computation of compound interest Tartaglia 
gave four methods, among them computation by steps 
from year to year, or computation with the aid of 
the formula b = ag**, although he does not give this 
formula. Computing of exchange was taught in its 
most simple form. It is said that bills of exchange 
were first used by the Jews who migrated into Lom- 
bardy after being driven from France in the seventh 
century. The Ghibellines who fled from Lombardy 
introduced exchange into Amsterdam, and from this 
city its use spread, f In 1445 letters of exchange were 
brought to Nuremberg. 

The chain rule {Kettensatz) ^ essentially an Indian 
method which is described by Brahmagupta, was de- 
veloped during the sixteenth century, but did not 
come into common use until two centuries later. The 
methods of notation differed. Pacioli and Tartaglia 

*Treutlein, DU deutacht Cost, SchlOmilch's 2^<V4cAr^/, Bd. 34, HI. A. 
t Unger, p. 90. 


wrote all numbers in a horizontal line and multiplied 
terms of even and of odd order into separate products. 
Stifel proceeded in the same manner, only he placed 
all terms vertically beneath ^one another. In the 
work of Rudolff, who also saw the advantage of can- 
cellation, we find the modern method of representing 
the chain rule, but the answer comes at the end.* 

About this time a new method of reckoning was 
introduced from Italy into Germany by the merchants, 
which came to occupy an important place in the six- 
teenth century, and still more so in the seventeenth. 
This Welsh (i. e., foreign) practice, as it soon came 
to be called, found its application in the development 
of the product of two terms of a proportion, especially 
when these were unlike quantities. The multiplier, 
together with the fraction belonging to it, was sepa- 
rated into its addends, to be derived successively one 
from another in the simplest possible manner. How 
well Stifel understood the real significance and appli- 
cability of the Welsh practice, the following statement 
shows :f **The Welsh practice is nothing more than 
a clever and entertaining discovery in the rule of three. 
But let him who is not acquainted with the Welsh 
practice rely upon the simple rule of three, and he 
will arrive at the same result which another obtains 
through the Welsh practice." At this time, too, we 
find tables of prices and tables of interest in use, 
their introduction being also ascribable to the Italians. 

* Unger, p. 92. . 1 Unger, p. 94. 


In the sixteenth century we also come upon examples 
for the reguia virginutn and the regula falsi in writings 
intended for elementary instruction in arithmetic, — 
writings into which, ordinarily, was introduced all the 
learning of the author. The significance of these 
rules, however, does not lie in the realm of elemen- 
tary arithmetic, but in that of equations. In the 
same way, a few arithmetic writings contained direc- 
tions for the construction of magic squares, and most 
of them also contained, as a side-issue, certain arith- 
metic puzzles and humorous questions (Rudolff calls 
them Schimpfrechnung), The latter are often mere 
disguises of algebraic equations (the problem of the 
hound and the hares, of the keg with three taps, of 
obtaining a number which has been changed by cer- 
tain operations, etc.). 

The seventeenth century brought essential innova- 
tions only in the province of commercial computation. 
While the sixteenth century was in possession of cor- 
rect methods in all computations of interest when 
the amount at the end of a given time was sought, 
there were usually grave blunders when the principal 
was to be obtained, that is, in computing the discount 
on a given sum. The discount in 100 was computed 
somewhat in this manner:* 100 dollars gives after 
two years 10 dollars in interest ; if one is to pay the 
100 dollars immediately, deduct 10 dollars." No less 
a man than Leibnitz pointed out that the discount 

*Unger, p. 132. 


must be reckoned upon 100. Among the majority of 
arithmeticians his method met with the misunder- 
standing that if the discount at 5% for one year is j/-^, 
the discount for two years must be /,-. It was not 
until the eighteenth century, after long and sharp 
controversy, that mathematicians and jurists united 
upon the correct formula. 

In the computation of exchange the Dutch were 
essentially in advance of other peoples. They pos- 
sessed special treatises in this line of commercial arith- 
metic and through them they were well acquainted 
with the fundamental principles of the arbitration of 
exchange. In the way of commercial arithmetic many 
expedients were discovered in the eighteenth century 
to aid in the performance of the fundamental opera- 
tions and in solving concrete problems. Calculation 
of exchange and arbitration of exchange were firmly 
established and thoroughly discussed by Clausberg. 
Especial consideration was given to what was called 
the Reesic rule, which was looked upon as differing 
from the well-known chain-rule. Rees's book, which 
was written in Dutch, was translated into French in 
1737, and from this language into German in 1739. 
In the construction of his series Rees began with the 
required term ; in the computation the elimination of 
fractions and cancellation came first, and then fol- 
lowed the remaining operations, multiplication and 

Computation of capital and interest was extended, 


through the establishment of insurance associations, , 
to a so-called political arithmetic, in which calcula- 
tion of contingencies and annuities held an important 

The first traces of conditions for the evolution of 
a political arithmetic'*' date back to the Roman prefect 
Ulpian, who about the opening of the third century 
A.D. projected a mortality table for Roman subjects, f 
But there are no traces among the Romans of life in- 
surance institutions proper. It is not until the Middle 
Ages that a few traces appear in the legal regulations 
of endowments and guild finances. From the four- 
teenth century there existed travel and accident in- 
surance companies which bound themselves, in con- 
sideration of the payment of a certain sum, to ransom 
the insured from captivity among the Turks or Moors. 

Among the guilds of the Middle Ages the idea of 
association for mutual assistance in fires, loss of cattle 
and similar losses had already assumed definite shape. 
To a still more marked degree was this the case among 
the guilds of artisans which arose after the Reforma- 
tion — guilds which established regular sick and burial 

We must consider tontines as the forerunner of 
annuity insurance. In the middle of the seventeenth 
century an Italian physician, Lorenzo Tonti, induced 
a number of persons in Paris to contribute sums of 

* Karup, Thtoretisches Handbuch der Lebensversicherung. 1871. 
t Cantor, I., p. 522. 


money the interest of which should be divided annu- 
ally among the surviving members. The French gov- 
ernment regarded this procedure as an easy method 
of obtaining money and established from 1689 to 1759 
ten state tontines which, however, were all given up 
in 1770, as it had been proved that this kind of state 
loan was not lucrative. 

In the meantime two steps had been taken which, 
by using the results of mathematical science, provided 
a secure foundation for the business of insurance. 
Pascal and Fermat had outlined the calculation of 
contingencies, and the Dutch statesman De Witt had 
made use of their methods to lay down in a separate 
treatise the principles of annuity insurance based upon 
the birth and death lists of several cities of Holland. 
On the other hand, Sir William Petty, in 1662, in a 
work on political arithmetic* contributed the first val- 
uable investigations concerning general mortality — a 
work which induced John Graunt to construct mor- 
tality tables. Mortality tables were also published by 
Kaspar Neumann, a Breslau clergyman, in 1692, and 
these attracted such attention that the Royal Society 
of London commissioned the astronomer Halley to 
verify these tables. With the aid of Neumann's ma- 
terial Halley constructed the first, complete tables of 
mortality for the various ages. Although these tables 
did not obtain the recognition they merited until half 
a century later, they furnished the foundation for all 

* Recently republished in inexpensive form in Cassell's National Library. 


later works of this kind, and hence Halley is justly 
called the inventor of mortality tables. 

The first modern life-insurance institutions were 
products of English enterprise. In the years 1698 and 
1699 th^re arose two unimportant companies whose 
field of operations remained limited. In the year 
1705, however, there appeared in London the ** Amic- 
able" which continued its corporate existence until 
1866. The ** Royal Exchange" and "London Assur- 
ance Corporation," two older associations for fire and 
marine insurance, included life insurance in their busi- 
ness in 1721, and are still in existence. There was soon 
felt among the managers of such institutions the im- 
perative need for reliable mortality tables, a fact which 
resulted in H alley's work being rescued from oblivion 
by Thomas Simpson, and in James Dodson's project- 
ing the first table of premiums, on a rising scale, after 
Halley's method. The oldest company which used 
as a basis these scientific innovations was the " Society 
for Equitable Assurances on Lives and Survivorships," 
founded in 1765. 

While at the beginning of the nineteenth century 
eight life insurance companies were already carrying 
on their beneficent work in England, there was at the 
same time not a single institution of this kind upon 
the Continent, in spite of the progress which had been 
made in the science of insurance by Leibnitz, the Ber- 
noullis, Euler and others. In France there appeared in 
1819 **La compagnie d' assurances g^ndrales sur la 


Vie. " In Bremen the founding of a life insurance com- 
pany was frustrated by the disturbances of the war in 
1806. It was not until 1828 that the two oldest Ger- 
man companies were formed, the one in Lubeck, the 
other in Gotha under the management of Ernst Wil- 
helm Arnoldi, the "Father of German Insurance." 

The nineteenth century has substantially enriched 
the literature of mortality tables, in such tables as 
those compiled by the Englishmen Arthur Morgan 
(in the eighteenth century) and Farr, by the Belgian 
Quetelet, and by the Germans, Brune, Heym, Fischer, 
Wittstein, and Scheffler. A. recent acquisition in this 
field is the table of deaths compiled in accordance 
with the vote of the international statistical congress 
at Budapest in 1876, which gives the mortality of the 
population of the German Empire for the ten years 
1871-1881. Further development and advancement 
of the science of insurance is provided for by the 
** Institute of Actuaries " founded in London in 1849 — 
an academic school with examinations in all branches 
of the subject. There has also been in Berlin since 
1868 a "College of the Science of Insurance," but 
it offers no opportunity for study and no examina- 

The following compilations furnish a survey of the 
conditions of insurance in the year 1890 and of its 
development in Germany.* There were in Germany: 

*Karap, Theoretisches Handbuch der Lebensversicherung^ 1871. Johnson, 
Universal Cyclopedia, under " Life-Insurance." 



at thk bkoinmimo mumbbr of number of fob trb bum 

of thk ykar life ins. persons in round numbers 

go's. insured (million marks) 

1852 IS 46.980 170 

1858 20 90,128 800 

1866 82 805,488 900 

1890 49 4250 

There were in 1890 : 



Germany 49 4250 million marks 

Great Britain and Ireland 75 900 " pounds 

France 17 8250 " francs 

Rest of Europe 58 8200 " francs 

United States of America 48 4000 " dollars 

All that the eighteenth century developed or dis- 
covered has been further advanced in the nineteenth. 
The center of gravity of practical calculation lies in 
commercial arithmetic. This is also finding expres- 
sion in an exceedingly rich literature which has been 
extended in an exhaustive manner in all its details, 
but which contains nothing essentially new except the 
methods of calculating interest in accounts current. 



T^HE beginnings of general mathematical science 
-*" are the first important outcome of special studies 
of number and magnitude ; they can be traced back 
to the earliest times, and their circle has only gradu- 
ally been expanded and completed. The first period 
reaches up to and includes the learning of the Arabs ; 
its contributions culminate in the complete solution 
of the quadratic equation of one unknown quantity, 
and in the trial method, chiefly by means of geometry, 
of solving equation^ of the third and fourth degrees. 
The second period includes the beginning of the 
development of the mathematical sciences among the 
peoples of the West from the eighth century to the 
middle of the seventeenth. The time of Gerbert forms 
the beginning and the time of Kepler the end of this 
period. Calculations with abstract quantities receive 
a material simplification in form through the use of 
abbreviated expressions for the development of for- 
mulae; the most important achievement lies in the 
purely algebraic solution of equations of the third and 
fourth degrees by means of radicals. 


The third period begins with Leibnitz and Newton 
and extends from the middle of the seventeenth cen- 
tury to the present time. In the first and larger part 
of this period a new light was diffused over fields 
which up to that time had been only partially ex- 
plored, by the discovery of the methods of higher 
analysis. At the end of this first epoch there appeared 
certain mathematicians who devoted themselves to 
the study of combinations but failed to reach the 
lofty points of view of a Leibnitz. Euler and La- 
grange, thereupon, assumed the leadership in the field 
of pure analysis. Euler led the way with more than 
seven hundred dissertations treating all branches of 
mathematics. The name of the great Gauss, who 
drew from the works of Newton and Euler the first 
nourishment for his creative genius, adorns the be- 
ginning of the second epoch of the third period. 
Through the publication of more than fifty large 
memoirs and a number of smaller ones, not alone on 
mathematical subjects but also on physics and astron- 
omy, he set in motion a multitude of impulses in the 
most varied directions. At this time, too, there opened 
new fields in which men like Abel, Jacobi, Cauchy, 
Dirichlet, Riemann, Weierstrass and others have made 
a series of most beautiful discoveries. 




I, General Arithmetic. 

However meagre the information which describes 
the evolution of mathematical knowledge among the 
earliest peoples, still we find isolated attempts among 
the Egyptians to express the fundamental processes 
by means of signs. In the earliest, mathematical pa- 
pyrus * we find as the sign of addition a pair of walk- 
ing legs travelling in the direction toward which the 
birds pictured are looking. The sign for subtraction 
consists of three parallel horizontal arrows. The sign 
for equality is ^. Computations are also to be found 
^ which show that the Egyptians were able to solve sim- 
ple problems in the field of arithmetic and geometric 
progressions. The last remark is true also of the 
Babylonians. They assumed that during the first five 
of the fifteen days between new moon and full moon, 
the gain in the lighted portion of its disc (which was 
divided into 240 parts) could be represented by a geo 
metric progression, during the ten following days by 
an arithmetic progression. Of the 240 parts there 
were visible on the first, second, third . . . fifteenth 

* Cantor, L, p. 37. 


5 10 20 40 1.20. 
1.36 1.52 2.08 2.24 2.40 
2.56 8.12 3.28 3.44 4. 

The system of notation is sexagesimal, so that we are 
to take 3. 28 = 3 X 60 + 28 = 208. * Besides th is there 
have been found on ancient Babylonian monuments 
the first sixty squares and the first thirty- two cubes 
in the sexagesimal system of notation. 

The spoils of Greek treasures are far richer. Even 
the name of the entire science ij fmOrffmriK-q comes from 
the Greek language. In the time of Plato the word 
fmOrffmra included all that was considered worthy of 
scientific instruction. It was not until the time of the 
Peripatetics, when the art of computation {logistic) 
and arithmetic, plane and solid geometry, astronomy 
and music were enumerated in the list of mathemat- 
ical sciences, that the word received its special signifi- 
cance. Especially with Heron of Alexandria logistic 
appears as elementary arithmetic, while arithmetic so 
called is a science involving the theory of numbers. 

Greek arithmetic and algebra appeared almost 
always under the guise of geometry, although the 
purely arithmetic and algebraic method of thinking 
was not altogether lacking, especially in later times. 
Aristotlef is familiar with the representation of quan- 
tities by letters of the alphabet, even when those 
quantities do not represent line-segments ; he says in 

•Cantor, I., p. 8i. t Cantor, I., p. 240. 

J I : -. / . 


one place : ** If A is the moving force, B that which is 
moved, T the distance, and A the time, etc. " By the 
time of Pappus there had already been developed a 
kind of reckoning with capital letters, since he was 
able to distinguish as many general quantities as there 
were such letters in the alphabet. (The small letters 
«! Pi y> stood for the numbers 1, 2, 3, . . .) Aristotle 
has a special word for ''continuous" and a definition 
for continuous quantities. Diophantus went farther 
than any of the other Greek writers. With him there 
already appear expressions for known and unknown 
, quantities. Hippocrates calls the square of a number 
Mvafujs (power), a word which was transferred to the 
Latin as potentia and obtained later its special mathe- 
matical significance. Diophantus gives particular 
names to all powers of unknown quantities up to the 
sixth, and introduces them in abbreviated forms, so 
that ^, ofi, x^, x^, x^y appear as 8^, ic^ 88^, 8k", kk^. 
The sign for known numbers is /jfi. In subtraction 
Diophantus makes use of the sign //t (an inverted and 
abridged tff) ; c, an abbreviation for laroi] equal, appears 
as the sign of equality. A term of an expression is 
called d8o$ ; this word went into Latin as species and 
was used in forming the title arithmetica speciosaz^Kl" 
gebra.* The formulae are usually given in words and 
are represented geometrically, as long as they have to 
do only with expressions of the second dimension. The 
first ten propositions in the second book of Euclid, 

*Cantpr, I., p. 44a. 


for example, are enunciations in words and geometric 
figures, and correspond among others to the expres- 
sions a{d-\- c-\- d, . ,)=ad-\-ac-\-ad-\- , (^H-^)^ 

= ^3 + 2ad + ^2 =(^a + d)a+(a + d)d. 

Geometry was with the Greeks also a meaiis for in- 
vestigations in the theory of numbers. This is seen, 
for instance, in the remarks concerning gnomon-num- 
bers. Among the Pythagoreans a square out of which 
a corner was cut in the shape of a square was called a 
gnomon. Euclid also used this expression for the 
figure A BCD£F which is obtained from the parallelo- 
gram ABCB' by cutting out the parallelogram DB^FE, 
The gnomon-number of the Pythagoreans is 2«+l; 
for when ABCB' is a square, the square upon DE = n 

can be made equal to the square on BC^= « -|- 1 by 
adding the square BE^\ X 1 and the rectangles AE 
= C^ = 1 X«> since we have «« + 2;^ + 1 = (« -|- 1)«. 
Expressions like plane and solid numbers used for 
the contents of spatial magnitudes of two and three 
dimensions also serve to indicate the constant tend- 


ency to objectify mathematical thought by means of 

All that was known concerning numbers up to the 
third century B. C. , Euclid comprehended in a general 
survey. In his Elements he speaks of magnitudes, with- 
out, however, explaining this concept, and he under- 
stands by this term, besides lines, angles, surfaces 
and solids, the natural numbers.* The difference be- 
tween even and odd, between prime and composite 
numbers, the method for finding the least common 
multiple and the greatest common divisor, the con- 
struction of rational right angled triangles according 
to Plato and the Pythagoreans — all these are familiar 
to him. A method (the "sieve") for sorting out 
prime numbers originated with Eratosthenes. It con- 
sists in writing down all the odd numbers from 3 
on, and then striking out all multiples of 3, 5, 7 . . . 
Diophantus stated that numbers of the form a^ + 2ab 
+ ^' represent a square and also that numbers of the 
form («' + ^') (^* + ^') can represent a sum of two 
squares in two ways ; for {ac -\- dd)* + {ad — dc)^ = 
(^ae — ddy + (ad+ bcf = {a^ -f b^) (^ + //«). 

The knowledge of the Greeks in the field of ele 
mentary series was quite comprehensive. The Pythag 
oreans began with the series of even and odd num- 
bers. The sum of the natural numbers gives the 
triangular number, the sum of the odd numbers the 
square, the sum of the even numbers gives the hetero- 



mecic (oblong) number of the form «(« + !). Square 
numbers they also recognised as the sum of two suc- 
cessive triangular numbers. The Neo- Pythagoreans 
and the Neo-Platonists made a study not only of po- 
lygonal but also of pyramidal numbers. Euclid treated 
geometrical progressions in his Elements. He ob- 
tained the sum of the series 1 -f 2 -f 4 + 8 . . . and 
noticed that when the sum of this series is a prime 
number, a ''perfect number" results from multiply- 
ing it by the last term of the series (1 + 2 + 4 = 7; 
7X4 = 28; 28 = 1 + 2 + 4+ 7+ 14; cf. p. 35). In- 
finite convergent series appear frequently in the works 
of Archimedes in the form of geometric series whose 
ratios are proper fractions ; for example, in calculating 
the area of the segment of a parabola, where the value 
of the series 1 + J + tV"I" • • • ^^ found to be ^. He 
also performs a number of calculations for obtaining 
the sum of an infinite series for the purpose of esti- 
mating areas and volumes. His methods are a sub- 
stitute for the modern methods of integration, which 
are used in cases of this kind, so that expressions like 

j xdx^^c^y I x^ dx'=^\(^ 

and other similar expressions are in their import and 
essence quite familiar to him.* 

The introduction of the irrational is to be traced 
back to Pythagoras, since he recognised that the hy- 

*Zeuthen, Die Lehre von den Kegelschnitten fm Alte-rtunt. Deutsch von 
V. Fischer-Benzon. 1886. 


potenuse of a right-angled isosceles triangle is in- 
commensurable with its sides. The Pythagorean 
Theodorus of Cyrene proved the irrationality of the 
square roots of 3, 5, 7, . . . 17.* 

Archytas classified numbers in general as rational 
and irrational. Euclid devoted to irrational quantities 
a particularly exhaustive investigation in his Ele- 
ments^ a work which belongs to the domain of Arith- 
metic as much as to that of Geometry. Three books 
among the thirteen, the seventh, eighth and ninth, 
are of purely arithmetic contents, and in the tenth 
book there appears a carefully wrought- out theory of 
** Incommensurable Quantities," that is, of irrational 
quantities, as well as a consideration of geometric 
ratios. At the end of this book Euclid shows in a 
very ingenious manner that the side of a square and 
its diagonal are incommensurable ; the demonstration 
culminates in the assertion that in the case of a ra- 
tional relationship between these two quantities a 
number must have at the same time the properties of 
an even and an odd number, f In his measurement 
of the circle Archimedes calculated quite a number of 
approximate values for square roots ; for example, 

1351 /^ 265 

Nothing definite, however, is known concerning the 

♦ Cantor, I., p. 170. 

t Montucla, I., p. ao8. Montucla says that he knew an architect who lived 
in the firm conviction that the square root of a could be represented as a 
ratio of finite integers, and who assured him that by this method he had 
already reached the xooth decimal. 


method he used. Heron also was acquainted with 

such approximate values (J instead of l/2, ^ instead 

of l/3) ;* and although he did not shrink from the 

labor of obtaining approximate values for square 

roots, in the majority of cases he contented himself 

. b 

with the well-known approximation l/a*±^=tfdb^, 

e. g., i/63 = l/8' — 1=8 — ^. Incase greater ex- 
actness was necessary, Heron f used the formula 
V a^ 4-T= « + T + ^ + ^+ • • • Incidentally he used 
the identity \/a^lf = al/J and asserted, for example, 
that 1/IO8 == i/P^ = 61/3 = 6-ff = 10 + ^ + ^. 
Moreover, we find in Heron*s Stereometrica the first 
example of the square root of a negative number, 
namely j/Sl — 144, which, however, without further 
consideration, is put down by the computer as 8 less 
•j^, which shows that negative quantities were un- 
known among the Greeks. It is true that Diophantus 
employed differences, but only those in which the 
minuend was greater than the subtrahend. Through 
Theon we are made acquainted with another method 
of extracting the square root ; it corresponds with the 
method in use at present, with the exception that the 
Babylonian sexagesimal fractions are used, as was 
customary until the introduction of decimal fractions. 
Furthermore, we find in Aristotle traces of the 
theory of combinations, and in Archimedes an at- 
tempt at the representation of a quantity which in- 

♦ Cantor, I., p. 368. t Tannery In Bordeaux Mim., IV., i88x. 


creases beyond all limits, first in his extension of the 
number-system, and then in his work entitled ^ofi 
lurqi (Latin arenarius, the sand-reckoner). Archi- 
medes arranges the first eight orders of the decimal 
system together in an octad ; 10^ octads constitute a 
period, and then these periods are arranged again 
according to the same law. In the sand-reckoning, 
Archimedes solves the problem of estimating the 
number of grains of sand that can be contained in a 
sphere which includes the whole universe. He as- 
sumes that 10,000 grains of sand take up the space of 
a poppy-seed, and he finds the sum of all the grains 
to be 10 000 000 units of the eighth period of his sys- 
tem, or 10^. It is possible that Archimedes in these 
observations intended to create a counterpart to the 
domain of infinitesimal quantities which appeared in 
his summations of series, a counterpart not accessible 
to the ordinary arithmetic. 

In the fragments with which we are acquainted 
from the writings of Roman surveyors {agrimensores) 
there are but few arithmetic portions, these having 
to do with polygonal and pyramidal numbers. Ob- 
viously they are of Greek origin, and the faulty style 
in parts proves that there was among the Romans no 
adequate comprehension of matters of this kind. 

The writings of the Hindu mathematicians are ex- 
ceedingly rich in matters of arithmetic. Their sym- 
bolism was quite highly developed at an early date. * 

* Cantor, I., p. 558. 


Aryabhatta calls the unknown quantity gulika ("little 
ball"), later yavattavat, or abbreviated j^a (**as much 
as"). The known quantity is called rupaka or ru 
(**coin"). If one quantity is to be added to another, 
it is placed after it without any particular sign. The 
same method is followed in subtraction, only in this 
case a dot is placed over the coefficient of the subtra- 
hend so that positive {dhana^ assets) and negative quan- 
tities (kshaya, liabilities) can be distinguished. The 
powers of a quantity also receive special designations. 
The second power is varga or va^ the third ghana or 
gha, the fourth va va, the fifth va gha ghata^ the sixth 
va gha, the seventh va va gha ghata {ghata signifies 
addition). The irrational square root is called karana 
or ka. In the ^ulvasutras, which are classed among 
the religious books of the Hindus, but which in addi- 
tion contain certain arithmetic and geometric deduc- 
tions, the word karana appears in conjunction with 
numerals; dvikaram^\/2, ^rikaramr=\/B, t/apakarani 
= 1/10. If several unknown quantities are to be dis- 
tinguished, the first is called ya ; the others are named 
after the colors : kalaka or ka (black) , nilaka or ni 
(blue), pitaka ox pi (yellow); for example, hy ya kabha 
is meant the quantity x'y, since bhavita or bha indi- 
cates multiplication. There is also a word for * * equal " ; 
but as a rule it is not used, since the mere placing of 
a number under another denotes their equality. 

In the extension of the domain of numbers to in- 
clude negative quantities the Hindus were certainly 


successful. They used them in their calculations, 
and obtained them as roots of equations, but never 
regarded them as proper solutions. Bhaskara was 
even aware that a square root can be both positive 
and negative, and also that V — a does not exist for 
the ordinary number-system. He says: "The square 
of a positive as well as of a negative number is posi- 
tive, and the square root of a positive number is 
double, positive, and negative. There can be no 
square root of a negative number, for this is no 
square. " * 

The fundamental operations of the Hindus, of 
which there were six, included raising to powers and 
extracting roots. In the extraction of square and cube 
roots Aryabhatta used the formulae for (a-f"^)* ^^^ 
{a + by, and he was aware of the advantage of sepa- 
rating the number into periods of two and three fig- 
ures each, respectively. Aryabhatta called the square 
root varga mula, and the cube root ghana mula {mula, 
root, used also of plants). Transformations of ex- 
pressions involving square roots were also known. 
Bhaskara applied the formulaf 

and was also able to reduce fractions with square roots 
in the denominator to forms having a rational denomi- 
nator. In some cases the approximation methods for 
square root closely resemble those of the Greeks. 

* Cantor, I., p. 585- t Cantor, I., p. 586. 


Problems in transpositions, of which only a few 
traces are found among the Greeks, occupy consider- 
able attention among the Indians. Bhaskara made 
use of formulae for permutations and combinations* 
with and without repetitions, and he was acquainted 
with quite a number of propositions involving the 
theory of numbers, which have reference to quadratic 
and cubic remainders as well as to rational right- 
angled triangles. But it is noticeable that we discover 
among the Indians nothing concerning perfect, ami- 
cable, defective, or redundant numbers. The knowl- 
edge of figurate numbers, which certain of the Greek 
schools cultivated with especial zeal, is likewise want- 
ing. On the contrary, we find in Aryabhatta, Brah- 
magupta and Bhaskara summations of arithmetic 
series, as well as of the series l* + 2*4-3*-f" • • •> ^^ 
4- 2* + 3* + • • /• The geometric series also appears in 

the works of Bhaskara. As regards calculation with 

a I 

zero, Bhaskara was aware that -^ = oo. ' 

The Chinese also show in their literature some 
traces of arithmetic investigations ; for example, the 
binomial coefficients for the first eight powers are 
given by Chu shi kih in the year 1303 as an "old 
method.** There is more to be found among the 
Arabs. Here we come at the outset upon the name of 
Al Khowarazmi, whose Algebra, which was probably 
translated into Latin by -^thelhard of Bath,f opens 

♦Cantor, I , p. 579. t Incorrect. See Biblioth. Math. (3) i: 520. 


with the words* **A1 Khowarazmi has spoken." In 
the Latin translation this name appears as Algoritmi, 
and to-day appears as algorism or algorithm^ a, word 
completely separated from all remembrance of Al Kho- 
warazmi, and much used for any method of computa- 
tion commonly employed and proceeding according 
to definite rules. In the beginning of the sixteenth 
century there appears in a published mathematical 
work a ^^philosophus nomine AlgorithmuSy^* a sufficient 
proof that the author knew the real meaning of the 
word algorism. But after this, all knowledge of the 
fact seems to disappear, and it was not until our own 
century that it was rediscovered by Reinaud and Bon- 

Al Khowarazmi increased his knowledge by study- 
ing the Greek and Indian models. A known quantity 
he calls a number, the unknown quantity Jidr (root) 
and its square mal (power). In Al Karkhi we find the 
expression kab (cube) for the third power, and there 
are formed from these expressions mal mal^^x^, mal 
kab = x^ , kab kab = x^, mal mal kab = x'^y etc. He also 
treats simple expressions with square roots, but with- 
out arriving at the results of the Hindus. There is a 
passage in Omar Khayyam from which it is to be in- 
ferred that the extraction of roots was always per- 
formed by the help of the formula for {a -(- by, Al 
Kalsadi J contributed something new by the introduc- 

* Cantor. I., p. 671. 

\ Jahrbuch Uier die Fortschritte der Mathematik^ 1887, p. 23. 

t Cantor, 1., p. 765. 


tion of a radical sign. Instead of placing the word 

jidr before the number of which the square root was 

to be extracted, as was the custom, Al Kalsadi makes 

use only of the initial letter •^ of this word and places 

it over the number, as, 


2 = 1/2; J2 = i/2|, 5 =2i/5. 

Among the Eastern Arabs the mathematicians 
who investigated the theory of numbers occupied 
themselves particularly with the attempt to discover 
rational right-angled triangles and with the problem 
of finding a square which, if increased or diminished 
by a given number, still gives a square. An anony- 
mous writer, for example, gave a portion of the the- 
ory of quadratic remainders, and Al Khojandi also 
demonstrated the proposition that upon the hypoth 
esis of rational numbers the sum of two cubes cannot 
be another third power.' There was also some knowl- 
edge of cubic remainders, as is seen in the applica- 
tion by Avicenna of the proof by excess of nines in 
the formation of powers. This mathematician gives 
propositions which can be briefly represented in the 

(9«=hl)« = l(w^^9), (9«db2)«=4(w^//9), 
(9«+l)« = (9// + 4)»=(9«+7)8 = l(w^^9), etc. 
Ibn al Banna has deductions of a similar kind which 
form the basis of a proof by eights and sevens. t 

In the domain of series the Arabs were acquainted 

♦Cantor, I., p. 712. t Cantor, I., p. 759. 


at least with arithmetic and geometric progressions 
and with the series of squares, cubes, and fourth 
powers. In this field Greek influence is unmistakable. 

2. Algebra, 

The work of Ahmes shows that the Egyptians 
were possessed of equations of the first degree, and 
used in their solution methods systematically chosen. 
The unknown x is called hau (heap); an equation* 
appears in the following form : heap, its f , its J, its 
4^, its whole, gives 37, that is \x -\- ^x -\- \x -{- x z=Z1 , . 

The ancient Greeks were acquainted with the so- 
lution of equations only in geometrical form. No- 
where, save in proportions, do we find developed ex- 
amples of equations of the first degree which would 
show unmistakably that the root of a linear equation 
with one unknown was ever determined by the inter- 
section of two straight lines ; but in the cases of equa- 
tions of the second and third degrees there is an 
abundance of material. In the matter of notation 
Diophantus makes the greatest advance. He calls 
the coefficients of the unknown quantity irX^dos. If 
there are several unknowns to be distinguished, he 
makes use of the ordinal numbers : 6 ir/ocoro? apt^/Aos, 6 
8€vre/oos, 6 rpiroq. An equation! appears in his works 
in the abbreviated form : 

* Matthiessen, GrundMii^^e der antiken und tnodernen Algebra der litiera' 
Un GUichungen, 1878, p. 269. Hereafter referred to as Matthiessen. 

t Matthiessen, p. 369. 



Diophantus classifies equations not according to the 
degree, but according to the number of essentially 
distinct terms. For this purpose he gives definite 
rules as to how equations can be brought to their sim- 
plest form, that is, the form in which both members 
of the equation have only positive terms. Practical 
problems which lead to equations of the first degree 
can be found in the works* of Archimedes and Heron; 
the latter gives some of the so-called ** fountain prob- 
lems," which remind one of certain passages in the 
work of Ahmes. Equations of the second degree 
were mostly in the form of proportions, and this 
method of operation in the domain of a geometric 
algebra was well known to the Greeks. They un- 
doubtedly understood how to represent by geometric 
figures equations of the form 

a' _ a' b' _ 

a a o 

where all quantities are linear. Every calculation of 
means in two equal ratios, i. e., in a proportion, was 
really nothing more than the solution of an equation. 
The Pythagorean school was acquainted with the 
arithmetic, the geometric and the harmonic means of 
two quantities; that is, they were able to solve geo- 
metrically the equations 

a-\'b ^ , 2iab 

x= — J5 — , x^ ^=aby x^= 

According to Nicomachus, Philolaus called the cube 



with its six surfaces, its eight corners, and its twelve 
edges, the geometric harmony, because it presented 
equal measurements in all directions ; from this fact, 
it is said, the terms "harmonic mean" and ** harmo- 
nic proportion *' were derived, the relationship being : 

12—8 12 ^ . 2-6-12 . 2ad 

-g— g = -g., whence 8= -^-^, i. e., x= -^-p. 

The number of distinct proportions was later in- 
creased to ten, although nothing essentially new was 
gained thereby. Euclid gives thorough analyses of 
proportions, that is, of the geometric solution of equa- 
tions of the first degree and of incomplete quadratics ; 
these, however, are not given as his own work, but as 
the result of the labors of Eudoxus. 

The solution of the equation of the second degree 
by the geometric method of applying areas, largely 
employed by the ancients, especially by Euclid, de- 
serves particular attention. 

In order to solve the equation 

by Euclid's method, the problem must first be put in 
the following form : 

A E B n 



/f G 


'*To the segment AB^^a apply the rectangle DH 
of known area = ^*, in such a way that CZT shall be a 
square." The figure shows that for CX=j, FH=i 
a:2 + 2;r • |- 4- (f ) ^ = ^3 _^ (f-) *; but by the Pythagorean 
proposition, ^ -f- (|-) * = ^*> whence EH=^ r = ^ -f jc, 
from which we have ;r=r — |-. The solution obtained 
by applying areas, in which case the square root is 
always regarded as positive, is accordingly nothing 
more than a constructive representation of the value 


In the same manner Euclid solves all equations of 

the form 

;r* ± ^x ± ^2 = 0, 

and he remarks in passing that where V V^ — (t)*' 
according to our notation, appears, the condition for 
a possible solution is ^>y. Negative quantities are 
nowhere considered ; but there is ground for inferring 
that in the case of two positive solutions the Greeks 
regarded both and that they also applied their method 
of solution to quadratic equations with numerical co- 
efficients.* By applying their knowledge of propor- 
tion, they were able to solve not only equations of the 
form x^ it d-.r zfc ^ = 0, but also of the more general 

for a as the ratio of two line-segments. Apollonius 

*Zeuthen, Die Lehr* mm den Kegelschnitten im Aitertum. Deutsch von 
V. Fischer-Benzon. i886. 


accomplished this with the aid of a conic, having the 

The Greeks were accordingly able to solve every gen- 
eral equation of the second degree having two essen- 
tially different coefficients, which might also contain 
numerical quantities, and to represent their positive 
roots geometrically. 

The three principal forms of equations of the sec- 
ond degree first to be freed from geometric statement 
and 'Completely solved, are 

s^-\'px=^q^ o^=.px-\-q^ px^^x^-^-q. 

The solution consisted in applying an area, the prob- 
lem being to apply to a given line a rectangle in such 
a manner that it would either contain a given area or 
be greater or less than this given area by a constant. 
For these three conditions there arose the technical 
expressions irapafioki^f vttc/ojSoXi;, lAAci^^is, which after 
Archimedes came to refer to conies.* 

In later times, with Heron and Diophantus, the 
solution of equations of the second degree was partly 
freed from the geometric representation, and passed 
into the form of an arithmetic computation proper 
(while disregarding the second sign in the square 

The equation of the third degree, owing to its 
dependence on geometric problems, played an im- 

♦ Tannery in Bordeaux Mint., IV. 



portant part among the Greeks. The problem of the 
duplication (and also the multiplication) of the cube 
attained especial celebrity. This problem demands 
nothing more than the solution of the continued pro- 
portion aix^=xiy=^y\2af that is, of the equation 
a^=z2ofi (in general a^ = ^(;fi). This problem is very 
old and was considered an especially important one 
by the leading Greek mathematicians. Of this we 
have evidence in a passage of Euripides in which he 
makes King Minos say concerning the tomb of Glau- 
cus which is to be rebuilt*: '* The enclosure is too 
small for a royal tomb : double it, but fail not in the 
cubical form." The numerous solutions of the equa- 
tion a^:=.2a^ obtained by Hippocrates, Plato, Me- 
naechmus, Archytas and others, followed the geomet- 
ric form, and in time the horizon was so considerably 
extended in this direction that Archimedes in the 
study of sections of a sphere solved equations of the 

by the intersection of two lines of the second degree, 
and in doing so also investigated the conditions to be 
fulfilled in order that there should be no root or two 
or three roots between and a. Since the method 
of reduction by means of which Archimedes obtains 
the equation o(^ — ax^ -^ b'^c^=(i can be applied with 
considerable ease to all torms of equations of the third 
degree, the merit of having set forth these equations 

* Cantor, I., p- 199. 


in a comprehensive manner and of having solved one 
of their principal groups by geometric methods be- 
longs without question to the Greeks.* 

We find the first trace of indeterminate equations 
in the cattle problem {Problema bovinum) of Archi-: 

This problem, which was published in the year 1778 by Les- 
sing, from a codex in the library at Wolfenbiittel, as the first of 
four unprinted fragments of Greek anthology, is given in twenty- 
two distichs. In all probability it originated directly with Archi- 
medes who desired to show by means of this example how, pro- 
ceeding from simple numerical quantities, one could easily arrive 
at very large numbers by the interweaving of conditions. The 
problem runs something as follows : f 

The sun had a herd of bulls and cows of different colors, (i) 
Of Bulls the white (^) were in number (i + J) of the black (A') 
and the yellow (F); the black (A') were (i-|-J) of the dappled (Z) 
and the yellow (F); the dappled (Z) were (i-|-f) of the white 
( W^ and the yellow ( F). (2) Of Cows which had the same colors 
(«;,x. >r, ^), z^=(i+4)(^+:r), :r = (i+J)(Z + ^). ^ = (4+i) 
(F+jv), jv=(J-|-f)(H^+it;). W-\-X is to be a square; F-f Z 
a triangular number. 

The problem presents nine equations with ten unknowns : 

^=(i + i)^+>^ ^=(i + J)Z + F 

Z = (i-f i)«^+F «, = (j + i)(^4.^) 

y =(j-^-^)(^-|-zt/) W-\-X—n^ 

Y-\-Z— ^— . 

*Zeuthen, Die Lehre von den Kegehchnitten im Altertum. Deutsch von 
V. Fischer-Benzon 1886. 

t Krumbiegel und Amthor, Das Problema bovinum des Archimedes. ScAlif- 
milch' s Zeitschrift , Bd. 25. HI. A.; Gow, p. 99. 



According to Amthor the solution is obtained by Pell's equation 
/' — 2-3 -7 '11 -29 -353 «2 = i, assuming the condition m = {mod. 
2 '4657), in which process there arises a continued fraction with a 
period of ninety-one convergents. If we omit the last two condi- 
tions, we get as the total number of cattle 5 916 837 175 686, a 
number which is nevertheless much smaller than that involved in 
the sand-reckoning of Archimedes. 

But the name of Diophantus is most closely con- 
nected with systems of equations of this kind. He 
endeavors to satisfy his indeterminate equations not 
by means of whole numbers, but merely by means of 

rational numbers (always excluding negative quanti- 

ties) of the form — where / and q must be positive in- 
tegers. It appears that Diophantus did not proceed 
in this field according to general methods, but rather 
by ingeniously following out special cases. At least 
those of his solutions of indeterminate equations of 
the first and second degrees with which we are ac- 
quainted permit of no other inference. Diophantus 
seems to have been not a little influenced by earlier 
works, such as those of Heron and Hypsicles. It may 
therefore be assumed that even before the Christian 
era there existed an indeterminate analysis upon 
which Diophantus could build.* 

The Hindu algebra reminds us in many respects 
of Diophantus and Heron. As in the case of Dio- 
phantus, the negative roots of an equation are not 
admitted as solutions, but they are consciously set 

♦p. Tannery, in Mimoires de Bordeaux, 1880. This view of Tannery's is 
controverted by Heath, T. L., Diophantos of Alexandria^ 1885, p. 135. 


aside, which marks an advance upon Diophantus. 
The transformation of equations, the combination of 
terms containing the same powers of the unknown, is 
also performed as in the works of Diophantus. The 
following is the representation of an equation accord- 
ing to Bhaskara :* 

va va 2 va 1 \ ru 30 

I, !• e. > 
ru 8 

2a:«— Jt-f 30 = 0jc« + 0a: + 8, or 2^2_^^30 = 8. 
Equations of the first degree appear not only with 
one, but also with several unknowns. The Hindu 
method of treating equations of the second degree 
shows material advance. In the first place, ax^ -\- bx 
= ^ is considered the only typef instead of the three 
Greek forms ax^ -^ dx = c, bx'\-c= ax^^ ax^ + ^ = ^•^• 
From this is easily derived AlO^o^ -\- ^abx^^ac, and 
then (2^j!jp + ^)' = 4dr^ -f ^2, whence it follows that 

Bhaskara goes still further. He considers both signs 
of the square root and also knows when it cannot be 
extracted. The two values of the root are, however, 
admitted by him as solutions only when both are posi- 
tive, — evidently because his quadratic equations ap- 
pear exclusively in connection with practical problems 
of geometric form. Bhaskara also solves equations 
of the third and fourth degrees in cases where these 

* Matthiessen, p. 269. tCaator, I., p. 585. 


equations can be reduced to equations of the second 
degree by means of advantageous transformations and 
the introduction of auxiliary quantities. 

The indeterminate analysis of the Hindus is espe- 
cially prominent. Here in contrast to Diophantus 
only solutions in positive integers are admitted. In- 
determinate equations of the first degree with two or 
more unknowns had already been solved by Arya- 
bhatta, and after him by Bhaskara, by a method in 
which the Euclidean algorism for finding the greatest 
common divisor is used ; so that the method of solu- 
tion corresponds at least in its fundamentals with the 
method of continued fractions. Indeterminate equa- 
tions of the second degree, for example those of the 
form xy = aX'\-by-\-c, are solved by arbitrarily as- 
signing a value to y and then obtaining x, or geo- 
metrically by the application of areas, or by a cyclic 
method.* This cyclic method does not necessarily 
lead to the desired end, but may nevertheless, by a 
skilful selection of auxiliary quantities, give integ- 
ral values. It consists in solving in the first place, 
instead of the equation ax^-\- b = cy^j the equation 
ax^ + 1 ^=)^^ This is done by the aid of the empiri- 
cally assumed equation aA^ -^B^^C^, from which 
other equations of the same form, aA^„-\- jB^=Ci, can 
be deduced by the solution of indeterminate equations 
of the first degree. By means of skilful combinations 

* Cantor, I., p. 591. 



the equations aA^f^ + B^=iCl furnish a solution of 

The algebra of the Chinese, at least in the earliest 
period, has this in common with the Greek, that equa- 
tions of the second degree are solved geometrically. 
In later times there appears to have been developed 
a method of approximation for determining the roots 
of higher algebraic equations. For the solution of in- 
determinate equations of the first degree the Chinese 
developed an independent method. It bears the name 
of the ** great expansion'* and its discovery is ascribed 
to Sun tse, who lived in the third century A. D. This 
method can best be briefly characterised by the fol- 
lowing example : Required a number x which when 
divided by 7, 11, 15 gives respectively the remainders 
2, 5, 7. Let kiy ^8, ^8» be found so that 

ll-l^>&i . , 15-7->^8 , , 
fj =^i + h — j-j - = ^2 + A» 

we have, for example, /'i=2, ^8 = 2, >tj = 8, and ob- 
tain the further results 

11 15-2 = 330, 330-2= 660, 

15- 7-2 = 210,* 210-5 = 1050, 

7-11-8 = 616, 616-7 = 4312, 

6022 247 

660 + 1050+4312 = 6022; --^^_ =5+ ^-^^-^; 

j|p=247 is then a solution of the given equation. f 

♦ Cantor, I., p. 593. t L. M atthiessen in Schldmilch' s Zettschrt/t, XXVI. 


In the writing of their equations the Chinese make 
as little use as the Hindus of a sign of equality. The 
positive coefficients were written in red, the negative 
in black. As a rule ide is placed beside the absolute 
term of the equation and yuen beside the coefficient of 
the first power ; the rest can be inferred from th.e ex- 
ample 14^ — 27ar = 17,* where r and b indicate the 
color of the coefficient : 

^14 or ^14 or ^14 

^00 ,00 ,00 

filyuen fi*J filyuen 

Xltde ;\ntde ,17. 

The Arabs were pupils of both the Hindus and 
the Greeks. They made use of the methods of their 
Greek and Hindu predecessors and developed them, 
especially in the direction of methods of calculation. 
Here we find the origin of the word algebra in the 
writings of Al Khowarazmi who, in the title of his 
work, speaks of ^^al-jabr wa'l muqabalahy^^ i. e., the 
science of redintegration and equation. This expres- 
sion denotes two of the principal operations used by 
the Arabs in the arrangement of equations. When 
from the equation a^-\-r = x^-{-px-\-r the new equa- 
tion a^=x^ +/* ^s formed, this is called al niuqabalah; 
the transformation which gives from the equation 
px — q-=^ofl the equation px^=x'^ -\- q^ a transforma- 
tion which was considered of great importance by the 

♦Cantor, I., p. 643. 


ancients, was called al-jabr, and this name was ex- 
tended to the science which deals in general with 

The earlier Arabs wrote out their equations in 
words, as for example, Al Khowarazmi"*" (in the Latin 
translation) : 

Census et quinque radices equantur viginti quatuor 
ar« + bx = 24; 

and Omar Khayyam, 

Cubus, latera et numerus aequales sunt quadratis 
«• -(- ^jc -|- c := ax^. 

In later times there arose among the Arabs quite an 
extended symbolism. This notation made the most 
marked progress among the Western Arabs. The 
unknown x was called j't'dr, its square mal; from the 
initials of these words they obtained the abbrevia- 
tions x=(J&, jc*= -u. Quantities which follow di- 
rectly one after another are added, but a special sign 
is used to denote subtraction. ''Equals" is denoted 
by the final letter of adala (equality), namely, by means 
of a final /am. In Al Kalsadif ^x^ = 12x + Q3 and 
^x*'\'X=7^ are represented by 

and thfe proportion 7 : 12 = 84 : ^ is given the form 

•^.-.84. -.12. -.7. 

^Matthie9«en, p. aCg. t Cantor, I., p. 767. 


Diophantus had already classified equations, not 
according to their degree, but according to the num- 
ber of their terms. This principle of classification we 
find completely developed among the Arabs. Ac- 
cording to this principle Al Khowarazmi* forms the 
following six groups for equations of the first and 
second degrees : 

cfl^=ax ("a square is equal to roots"), 
^=a ("a square is equal to a constant*'), 
ax = bf a^ -\- ax = b, or* + a = bx, ax-^ b = ^9 
(** roots and a constant are equal to a square"). 

The Arabs knew how to solve equations of the first 
degree by four different methods, only one of which 
has particular interest, and that because in modem 
algebra it has been developed as a method of approx- 
imation for equations of higher degree. This method 
of solution, Hindu in its origin, is found in particular 
in Ibn al Banna and Al Kalsadi and is there called 
the method of the scales. It went over into the Latin 
translations as the regula falsorum and reguia falsi. To 
illustrate, let the equation ajc-|-3 = be givenf and 
let z\ and z% be any numerical quantities ; then if we 
place azx + b =yi, az^ + b =j^8, 

x= , 


as can readily be seen. Ibn al Banna makes use of 
the following graphic plan for the calculation of the 
value of x : 

* Matthiessen. p. 270. t Matthiessen, p. 277. 






The geometric representation, which with ^ as a neg- 
ative quantity somewhat resembles a pair of scales, 
would be as follows, letting OB\=zi, 0B2 = Zi, BiC\ 
=yu B^C^=y^, OA=x\ 

From this there results directly 

x — z% y^^ 

that is, that the errors in the substitutions bear to 
each other the same ratio as the errors in the results, 
the method apparently being discovered through geo- 
metric considerations. 

In the case of equations of the second degree Al 
Khowarazmi gives in the first place a purely mechan- 
ical solution (negative roots being recognised but not 
admitted), and then a proof by means of a geometric 
figure. He also undertakes an investigation of the 
number of solutions. In the case of 

3^-\- c^=^bx, from which ar=^d=l/(4)' — c^ 

Al Khowarazmi obtains two solutions, one or none 
according as 



He gives the geometric proof for the correctness of 
the solution of an equation like jip>-f 2jc=15, where 
he takes ^ = 3, in two forms, either by means of a 
perfectly symmetric figure, or by the gnomon. In 
the first case, lor AB=X9 BC=i, BD=lf we have 





C D 

^ + 4-i-« + 4-(i)« = 15+l, (ji: + l)« = 16; in the 
second we have jc* + 2 • 1 • jp -f 1' = 15 -|- 1. In the 
treatment of equations of the form aafi^ d= bx"" ib r = 0, 
the theory of quadratic equations receives still further 
development at the hands of Al Kalsadi. 

Equations of higher degree than the second, in 
the form in which they presented themselves to the 
Arabs in the geometric or stereometric problems of 
the Greek type, were not solved by them arithmeti- 
cally, but only by geometric methods with the aid 
of conies. Here Omar Khayyam* proceeded most 
systematically. He solved the following equations 
of the third degree geometrically : 

* Matthiessen, pp. 367, 894, 945. 


qx = ^, o^-\- qx =px^ , x^ ±: px^=^ r, x^ ± /^'+ r^=qXf 
px^=2a^, ^±^ar = r, ;c*+r=/^*, c^-±,p3^^=qx-{-r. 

The following is the method of expression which he 
employs in these cases : 

'*A cube and square are equal to roots;" 
''a cube is equal to roots, squares and one number," 
when the equations 

a^-{-px^^=qx, c^ = pcfl '\' qx '\' r 

are to be expressed. Omar calls all binomial forms 
simple equations ; trinomial and quadrinomial forms 
he calls composite equations. He was unable to solve 
the latter, even by geometric methods, in case they 
reached the fourth degree. 

The indeterminate analysis of the Arabs must be 
traced back to Diophantus. In the solution of inde- 
terminate equations of the first and second degree 
Al Karkhi gives integral and fractional numbers, like 
Diophantus, and excludes only irrational quantities. 
The Arabs were familiar with a number of proposi- 
tions in regard to Pythagorean triangles without hav- 
ing investigated this field in a thoroughly systematic 



As long as the cultivation of the sciences among 
the Western peoples was almost entirely confined to 



the monasteries, during a period lasting from the 
eighth century to the twelfth, no evidence appeared 
of any progress in the general theory of numbers. 
As in the learned Roman world after the end of the 
fifth century, so now men recognised seven liberal 
arts, — the trivium, embracing grammar, rhetoric and 
dialectics, and the quadrivium^ embracing arithmetic, 
geometry, music and astronomy. * But through Arab 
influence, operating in part directly and in part 
through writings, there followed in Italy and later also 
in France and Germany a golden age of mathemat- 
ical activity whose influence is prominent in all the 
literature of that time. Thus Dante, in the fourth 
canto of the Divina Commedia mentions among the 

"... who slow their eyes around 
Majestically moved, and in their port 
Bore eminent authority," 

a Euclid, a Ptolemy, a Hippocrates and an Avicenna. 
There also arose, as a further development of cer- 
tain famous cloister, cathedral and chapter schools, 
and in rare instances, independent of them, the first 
universities, at Paris, Oxford, Bologna, and Cam- 
bridge, which in the course of the twelfth century 
associated the separate faculties, and from the begin- 
ning of the thirteenth century became famous as Stu- 
dia generalia,'\ Before long universities were also es- 

*Muller, Hisiorisch-€tymologitche Studien ^er mathematische Termino- 
togie, 1887. 

t Suter, Die Mathematik aufden Universitaten des MitUlaltert^ 1887. 


tablished in Germany (Prague, 1348; Vienna, 1365; 
Heidelberg, 1386 ; Cologne, 1388 ; Erfurt, 1392 ; Leip- 
zig, 1409; Rostock, 1419; Greifswalde, 1456; Basel, 
1459; Ingolstadt, 1472; Tubingen and Mainz, 1477), 
in which for a long while mathematical instruction 
constituted merely an appendage to philosophical re- 
search. We must look upon Johann von Gmunden as 
the first professor in a German university to devote 
himself exclusively to the department of mathematics. 
From the year 1420 he lectured in Vienna upon mathe- 
matical branches only, and no longer upon all depart- 
ments of philosophy, a practice which was then uni- 

z. General Arithmetic, 

Even Fibonacci made use of words to express 
mathematical rules, or represented them by means of 
line-segments. On the other hand, we find that Luca 
Pacioli, who was far inferior to his predecessor in 
arithmetic inventiveness, used the abbreviations ./., 
,m.y R, for plus, minus, and radix (root). As early as 
1484, ten years before Pacioli, Nicolas Chuquet had 
written a work, in all probability based upon the re- 
searches of Oresme, in which there appear not only 
the signs / and in (for plus and minus), but also ex- 
pressions like 

IJMO, 5M7 for Ho, t/17. 
He also used the Cartesian exponent-notation, and 


the expressions eguipolence, equipolent^ for equivalence 
and equivalent.* 

Distinctively symbolic arithmetic was developed 
upon German soil. In German general arithmetic 
and algebra, in the Deutsche Coss, the symbols + and 
— for plus and minus are characteristic, f They were 
in common use while the Italian school was still writ- 
ing / and in. The earliest known appearance of these 
signs is in a manuscript {Regula Cose vel Algebre') of 
the Vienna library, dating from the middle of the fif- 
teenth century. In the beginning of the seventeenth 
century Reymers and Faulhaber used the sign -4-, J 
and Peter Roth the sign -h- as minus signs. 

Among the Italians of the thirteenth and four- 
teenth centuries, in imitation of the Arabs, the course 
of an arithmetic operation was expressed entirely in 
words. Nevertheless, abbreviations were gradually in- 
troduced and Luca Pacioli was acquainted with such 
abbreviations to express the first twenty-nine powers 
of the unknown quantity. In his treatise the absolute 
term and jp, x^^ y^^ x^, x^, x^, : . . are always respec- 
tively represented by numero or «% cosa or co^ censo or 
cey cubo or cu^ censo de censo or ce . ce^ pritno relato or 
^.*r*, censo de cuba or . . . 

The Germans made use of symbols of their own 

*A. Marre in Boncompagni's BulUttno, XIII. Jahrbuch liber die Fort- 
Khritte der Math., 1881, p. 8. 

tTreutlein, "Die deutsche Coss," SchlSmtlch** Zeitschrift, Bd. 24, HI. A. 
Hereafter referred to as Treutlein. 

t The sign -1- is first used as a sign of division in Rahn's Teutscht Algebrm 
Ziirich, 1659. 


invention. Rudolfi and Riese represented the abso- 
lute term and the powers of the unknown quantity in 
the following manner : Dragma^ abbreviated in writ- 
ing, ^\ radix (or coss^ i. e., root of the equation) is 
expressed by a sign resembling an r with a little flour- 
ish ; zensus by j ; cubus by r with a lofig flourish on 
top in the shape of ^n / (in the following pages this 
will be represented merely by c)\ zensus de zensu (zens- 
dezens) by 55, sursolidum by )8 or § ; zensikubus by y ; 
bissursolidum by \>\% or Bg; zensus zensui de zensu (zens- 
zensdezens) by 555 ; cubus de cubo by cc. 

There are two opinions concerning the origin of 
the X of mathematicians. According to the one, it was 
originally an r {radix) written with a flourish which 
gradually came to resemble an x^ while the original 
meaning was forgotten. Descartes, in the seventeenth 
century, first used the x in our present sense.* The 
other explanation depends upon the fact that it is cus- 
tomary in Spain to represent an Arabic j by a Latin 
X where whole words and sentences are in question ; 
for instance the quantity 12^;, in Arabic (/* is repre- 
sented by 12 xai^ more correctly by 12 sai. Accord- 
ing to this view, the x of the mathematicians would 
be an abbreviation of the Arabic sai = xai, but this 
theory is now abandoned. 

By the older cossistsf these abbreviations are in- 
troduced without any explanation ', Stifel, however, 

*Treutlein. G. Wertheim in SchlSmiUk't ZtitschH/t, Bd. 44> HI. A. 


considers it necessary to give his readers suitable ex- 
planations. The word " root," used for the first power 
of an unknown quantity he explains by means of 
the geometric progression, << because all successive 
members of the series develop from the first as from 
a root " ; he puts for x^, x\ x^, x^, jp*, . , . the signs 1, 
Ix, Ij, Ic, I55, . . . and calls these **cossic numbers,** 
which can be continued to infinity, while to each is 
assigned a definite order-number, that is, an expo- 
nent. In the German edition of Rudolff*s Coss, Stifel 
at first writes the *<cossic series" to the seventeenth 
power in the manner already indicated, but also later 
as follows : 

1 . 12t. 12t2t. 1717171. ^7X7X71U. etc. 

He also makes use of the letters B and (£ in writing 
this expression. The nearest approach to our present 
notation is to be found in Burgi and Reymers, where 
with the aid of '* exponents*' or ''characteristics'* the 
polynomial Sx^ + 12a:« — 9ar* + 10^8 + 3^2 ^ 7^ _ 4 jg 
represented in the following manner : 


8-1-12 — 9 + 10 + 3 -f 7—4 
In Scheubel we find for x, x^, x^, x^, x'^ . . , , pri,^ 
sec.y ter,, guar,, quin.y and in Ramus /, q, c, bq, s as 
abbreviations for latus^ quadratuSy cubus, biquadratus, 

The product (Ix^ _ 3je + 2) (5:r — 3) == 35jc» — 36:r» 
+ 19^ — 6 is represented in its development by Gram- 
mateus, Stifel, and Ramus in the following manner : 



by hpru—ZN hx —3 

35/<?r. — ISjc.+ lO/r/. 35<:— 155 + lOjc 

— 21JC.+ ^firi.—QN — 2I5+ 9ar— 6 

35/^r.— 36jir.+ 19/r/. — 6^ 35r— 365+ 19*— 6 


7^_ 3/+ 2 
5/— 3 

35^— 15<jr+10/ 
— 2V+ 9/— 6 

35^r_36^ + 19/— 6. 

As early as the fifteenth century the German Coss 
made use of a special symbol to indicate the extrac- 
tion of the root. At first .4 was used for l/4 ; this 
period placed before the number was soon extended 
by means of a stroke appended to it. Riese and 
Rudolff write merely |/4 for l/4. Stifel takes the 
first step towards a more general comprehension of 
radical quantities in his Arithmeiica integra^ where the 
second, third, fourth, fifth, roots of six are represented 
^y l/$^> V^^i l/$5^» V%^^ while elsewhere the symbols 

are used as radical signs. These symbols, of which 
the first two occur in Rudolff and the other three in a 
work of Stifel, indicate respectively the third, fourth, 
second, third, and fourth roots of the numbers which 
they precede. 


Rudolff gives a few rules for operations with rad- 
ical quantities, but without demonstrations. Like 
Fibonacci he calls an irrational number a numerus 
surdus. Such expressions as the following are intro- 
duced : 

l/a^c + Vb^c = V{a + bf • c, 

X _x {y'a-^V~b) 

\/adr\/b a — b 

Stifel enters upon the subject of irrational numbers 
with especial zeal and even refers to the speculations 
of Euclid, but preserves in all his developments a 
well-grounded independence. Stifel distinguishes two 
classes of irrational numbers : principal and subordi- 
nate (^Haupt- und Nebenarteti), In the first class are 
included (1) simple irrational numbers of the form 
v^, (2) binomial irrationals with the positive sign, as 

1/35^0 + V^. 4+1/56, |/3l2 + y'a2, 

(3) square roots of such binomial irrationals as 

Vl • 1/56 + 1/58 = ^ 1/6 + 1/ 8; 

1/3-5 + 1/55 = "^5 + V^» 

(4) binomial irrationals with the negative sign, as 
1/5 jlO — 1/556, and (5) square roots of such binomial 
irrationals, as 

' t/$-l/56-i/58 = ^l/6-i/8. 

The subordinate class of irrational quantities, accord- 
ing to Stifel, includes expressions like 


V7? + T/33 + 1/55, T/52 + 1/534 + |/r3, 
1/55 • 1/ 36 + 2.- . i/5^jA8+j/55l2 

=i^i/6+2-i!/e/8 + e/i2 

Fibonacci evidently obtained his knowledge of 
negative quantities from the Arabs, and like them he 
does not admit negative quantities as the roots of an 
equation. Pacioli enunciates the rule, minus times 
minus gives always plus, but he makes use of it only 
for the expansion of expressions of the form {p — q) 
(r — s). Cardan proceeds in the same way; he recog- 
nises negative roots of an equation, but he calls them 
aestimationes falsae 01 fictae* and attaches to them no 
independent significance. Stifel calls negative quan- 
tities numeri absurdi. Harriot is the first to consider 
negative quantities in themselves, allowing them to 
form one side of an equation. Calculations involving 
negative quantities consequently do not begin until 
the seventeenth century. It is the same with irratio- 
nal numbers ; Stifel is the first to include them among 
numbers proper. 

Imaginary quantities are scarcely mentioned. Car- 
dan incidentally proves that 

(5 + 1/IZT5)- (5— 1/^=^15) =40. . 

Bombelli goes considerably farther. Although not 
entering into the nature of imaginary quantities, of 
which he calls -\-\/ — 1 piu di tneno, and — 1/ — 1 
tneno di menoj he gives rules for the treatment of ex- 

*Ars magna^ 1545. Cap. I., 6. 


pressions of the form a-\'b V — 1, as they occur in 
the solution of the cubic equation. 

The Italian school early made considerable ad- 
vancement in calculations involving powers. Nicole 
Oresme* had long since instituted calculations with 
fractional exponents. In his notation 

it appears that he was familiar with the formulae 

In the transformation of roots Cardan made the first 
important advance by writing 

V a + V~b=p + Vq, V a — V~b=^p— V~q 

and therefore '^ a^ — b =/* — ^ = ^, «* — b = c^, Bom- 
bellif enlarged upon this observation and wrote 

f^a + \/~^b=p + l/^, ^a — V^^ =p — l/^, 
from which follows v a^ + b =/2 _|_ ^. With reference 
to the equation jc'=:15:r-(-4 he discovered that 

= 2 + t/I^ -f 2 — l/ITi =4. 
For in this case 

j)« + ^ = 5, (;>+i/II^)»=:2 + i/=12i, 
(/— 1/=^)' = 2 — i/irT2l, 

become through addition /* — 3/^ = 2, and with ^= 
5 — /*, 4/8 — 15/ = 2, and consequently (by trial)/=2 
and ^=cl 

* Hankel, p. 350. t Cantor, II., p. 572. 


The extraction of square and cube roots accord- 
ing to the Arab, or rather the Indian, method, was set 
forth by Grammateus. In the process of extracting 
the square root, for the purpose of dividing the num- 
ber into periods, points are placed over the first, third, 
fifth, etc., figures, counting from right to left. Stifel* 
developed the extracting of roots to a considerable 
extent ; it is undoubtedly for this purpose that he 
worked out a table of binomial coefficients as far as 
(a-\-dy, in which, for example, the line for (« + *)* 
reads : 

Ijj .4 6 4 1 . 

The theory of series in this period made no ad- 
vance upon the knowledge of the Arabs. Peurbach 
found the sum of the arithmetic and the geometric 
progressions. Stifel examined the series of natural 
numbers, of even and of odd numbers and deduced 
from them certain power series. In regard to these 
series he was familiar, through Cardan, with the the- 
orem that 1 + 2 + 2« + 2» + . . + 2— 1 = 2*— 1. With 
Stifel geometric progressions appear in an application 
which is not found in Euclid's treatment of means. f 
As is well known, n geometric means are inserted be- 
tween the two quantities a and d by means of the 

— = — — —= =^^=^=^ 

Xl X2 Xz x^ b 

* Cantor, II., pp. 397, 409. tTreatlein. 




where q:=**^l/±. Stifel inserts five geometric means 
between the numbers 6 and 18 in the following man- 
ner : 

1 8 9 27 81 248 729 

\/Y^ \/y^ \/y^ \/3^7 \/3^81 y/yfl^Z \/3^29 

6 v'a^lSOOeS v/r648 v'slOS ^/tl944 v/3cll337408 18 

in which the last line is obtained from the preceding 
by multiplying by 6. Stifel makes use of this solution 
for the purpose of duplicating the cube. He selects 6 
for the edge of the given cube ; three geometric means 
are to be inserted between 6 and 12, and as ^=i?^^, 
the edge of the required cube will be x^=6f^2 = 
]/ c432. This length is constructed geometrically by 
Stifel in the following manner : 

In the right angled triangle ACB, with the hypotenuse 
BC, let AB = 6, AC=12; make AD = DC, AE = 
ED, AF=FE, F/=/E, /K=/C=/L, Then AK 
is the first, AL the second geometric mean between 
6 and 12. This construction, which Stifel regards as 
entirely correct, is only an approximation, since 
^A'^ 7. 5 instead of 61^2 = 7.56, ^Z = 3l/T0 = 9.487 
instead of 6#^=: 9 . 524. 



Simple facts involving the theory of numbers were 
also known to Stifel, such as theorems relating to 
perfect and diametral numbers and to magic squares. 

A diametral number is the product of two numbers 
the sum of whose squares is a rational square, the 
square of the diameter, e. g., 65« = 252 + 6O2 == 39« + 
52«, and hence 25.60 = 1500 and 39.52 = 2028 are 
diametral numbers of equal diameter. 

























Magic squares are figures resembling a chess 
board, in which the terms of an arithmetic progres- 
sion are so arranged that their sum, whether taken 
diagonally or by rows or columns, is always the same. 
A magic square containing an odd number of cells, 
which is easier to construct than one containing an 
even number, can be obtained in the following man- 
ner : Place 1 in the cell beneath the central one, and 
the other numbers, in their natural order, in the empty 
cells in a diagonal direction. Upon coming to a cell 


already occupied, pass vertically downwards over two 
cells.* Possibly magic squares were known to the Hin- 
dus ; but of this there is no certain evidence. "f Manuel 
MoschopulusJ (probably in the fourteenth century) 
touched upon the subject of magic squares. He gave 
definite rules for the construction of these figures, 
which long after found a wider diffusion through La- 
hire and MoUweide. During the Middle Ages magic 
squares formed a part of the wide-spread number- 
mysticism. Stifel was the first to investigate them in 
a scientific way, although Adam Riese had already in- 
troduced the subject into Germany, but neither he nor 
Riese was able to give a simple rule for their con- 
struction. We may nevertheless assume that towards 
the end of the sixteenth century such rules were known 
to a few German mathematicians, § as for instance, to 
the Rechenmeister of Nuremberg, Peter Roth. In the 
year 1612 Bachet published in his Probletnes plaisants |1 
a general rule for squares containing^to odd number 
of cells, but acknowledged that he had not succeeded 
in finding a solution for squares containing an even 
number. Fr^nicle was the first to make a real ad- 
vance beyond Bachet. He gave rules (1693) for both 
classes of squares, and even discovered squares that 
maintain their characteristics after striking off th 

• Unger, p. 109. 

t Montucla, Htstotre des Mathtmatiques^ 1799-1802. 
X Cantor, I., p. 480. 

IGiesing, Leben und Schriften Leonardo's da Pisa, 1886. 
I This work is now accessible in a new edition published in 1884, Paris. 


outer rows and columns. In 1816 Mollweide collected 
the scattered rules into a book, De quadratis magicis, 
which is distinguished by its simplicity and scientific 
form. More modem works are due to Hugel (Ans- 
bach, 1859), to Pessl (Amberg, 1872), who also con- 
siders a magic cylinder, and to Thompson (^Quarterly 
Journal of Mathematics, Vol. X. ), by whose rules the 
magic square with the side pn is deduced from the 
square with the side n,^ 

2. Algebra. 

Towards the end of the Middle Ages the Ars major. 
Arte maggiore. Algebra or the Coss is opposed to the 
ordinary arithmetic {Ars minor). The Italians called 
the theory of equations either simply Algebra, like the 
Arabs, or Ars magna, Ars ret et census (very common 
after the time of Leonardo and fully settled in Regio- 
montanus), La regola della cosa {cosa = reSy thing) , 
Ars cossica or Regula cosae. The German algebraists 
of the fifteenth and sixteenth centuries called it Coss, 
Regula Coss, Algebra, or, like the Greeks, Logistic. 
Vieta used the term Arithmetica speciosa, and Reymers 
Arithmetica analytica, giving the section treating of 
equations the special title von der Aequation, The 
method of representing equations gradually took on 
the modern form. Equality was generally, even by 
the cossists, expressed by words ; it was not until the 
middle of the seventeenth century that a special sym- 

*G8ntber, "Ueber magische Quadrate," Grunert'sArcA., Bd. 57. 


bol came into common use. The following are exam- 
ples of the different methods of representing equa- 
tions :* 
Cardan : 

Cubus / 6 rebus aequalis 20, ^ -f- 6jr = 20 ; 

Vieta : 

1(7— 8^+16iV^ aequ.40, x^ — Sx^ + lQx = 4tO; 

Regiomontanus : 

16 census et 2000 aequ. 680 rebus, 16jt« + 2000 = 

Reymers : 


l^r 65532 +18 -^30 -5-18 +12 -^8; 
x^^ = 6bb32x^^+lSx^^ — 30x^ — lSx^ + 12xS; 

Descartes : 

z^oo az — dd z^=:iaz — ^* ; 

y _ 8/ — 1^^ + 8r*x 0, y — Sji^—y^ + 8^^= 

^6 * * * * — dx x> 0, x^ — ^x =0 

jp6 ♦ * * * _^ 30 0, x'^ — d =0 

Hudde : 

x^zogx.r,' x^^=qx-\-r. 

In Euler's time the last transformation in the develop 
ment of the modern form had already been accom- 

Equations of the first degree offer no occasion for 
remark. We may nevertheless call attention to the 
peculiar form of the proportion which is found in 
Grammateus and Apian. f The former writes : *' Wie 

* Matthiessen, GrundnUg^e der antiken und modemtn Algebra^ 2 ed., 1896, 
p. 270, etc. 

t Gerhardt. Gtschichte der Mathemaiik in Deuttckland^ 1877. 


sich hadt a zum b^ also hat sich c zum <//' and the 
latter places 

4-12-9-0 for ^ = |. 

Leonardo of Pisa solved equations of the second de- 
gree in identically the same way as the Arabs.* Car- 
dan recognized two roots of a quadratic equation, even 
when one of them was negative; but he did not regard 
such a root as forming an actual solution. Rudolfi 
recognized only positive roots, and Stifel stated ex- 
plicitly that, with the exception of the case of quad- 
ratic equations with two positive roots, no equation 
can have more than one root. In general, the solu- 
tion was affected in the manner laid down by Gram- 
mateusf in the example 12jp-|- 24 = 2 J^^*: ''Proceed 
thus: divide 24iV by 2J^ sec, which gives lOfdf 
(10f=:a). Also divide 12 pri. by 2J^ sec, which 
gives the result 5^/^(5^ = ^). Square the half of ^, 
which gives ^W"» ^^ which add d5 = 10|, giving ^^^, 
of which the square root is W. Add this to \ of b^ or 
^|, and 7 is the number represented by 1 pri. Proof : 
12x7A^=84A^; add 24iV^, =\^%N. 2J$ sec multi- 
plied by 49 must also give 108A^." 

This ''German Coss " was certainly cultivated by 
Hans Bernecker in Leipzig and by Hans Conrad in 
EislebenJ (about 1525), yet no memoranda by either 
of these mathematicians have been found. The Uni- 
versity of Vienna encouraged Grammateus to publish, 

♦ Cantor, II., p. 31. t Gerhardt. X Cantor, II., p. 387. 


in the year 1523, the first German treatise on Algebra 
under the title, ^^ Eyn new kunstlich behend vnd gewiss 
Rechenbuchlin | vff alle Kauffmannschafft, Nach Ge- 
meynen Regeln de tre. Welschen practic. Regeln 
falsi. Etlichen Regeln Cosse . . Buchhalten . . Visier 
Ruthen zu machen." Adam Riese, who had pub- 
lished his Arithmetic in 1518, completed in 1524 the 
manuscript of the Coss ; but it remained in manu- 
script and was not found until 1855 in Marienberg. 
The Coss published by Christoff Rudolff in 1525 in 
Strassburg met with universal favor. This work, 
which is provided with many examples, all completely 
solved, is described in the following words : 

' ' Behend vnd HQbsch Rechnung durch die kunstreichen re- 
geln Algebre | so gemeinicklich die Coss genennt werden. Darinnen 
alles so treulich an Tag geben | das auch allein ausz vleissigem 
Lesen on alien mUndtliche vnterricht mag begriffen werden. Hind- 
angesetzt die meinung aller dere | so bisher vil vngegrundten regeln 
angehangen. Einem jeden liebhaber diser kunst lustig vnd ergetz- 
lich Zusamen bracht durdh Cbristoffen Rudolff von Jawer."* 

The principal work of the German Coss is Michael 
StifePs Arithmetica Integra, published in Nuremberg in 
1544. In this book, besides the more common opera- 
tions of arithmetic, not only are irrational quantities 
treated at length, but there are also to be found appli- 

*A translation would read somewhat as follows: "Rapid and neat com- 
putation by means of the ingenious rules of algebra, commonly designated 
the Coss. Wherein are faithfully elucidated all things in such wise that they 
may be comprehended from diligent reading alone, without any oral instruc- 
tion whatsoever. In disregard of the opinions of all those who hitherto have 
adhered to numerous unfounded rules. Happily and divertingly collected 
ior lovers of this art, by Christoff Rudolff, of Jauer." 


cations of algebra to geometry. Stifel also published 
in 1553 Die Cost Christoffs Rudolffs mtt schonen Ex- 
empeln der Cosz Gebessert vnd sehr gemehrt^ with copi- 
ous appendices of his own, giving compendia of the 
Coss. With pardonable self -appreciation Stifel as- 
serts, "It is my purpose in such matters (as far as I 
am able) from complexity to produce simplicity. 
Therefore from many rules of the Coss I have formed 
a single rule and from the many methods for roots 
have also established one uniform method for the in- 
numerable cases." 

StifePs writings were laid under great contribu- 
tion by later writers on mathematics in widely distant 
lands, usually with no mention of his name. This 
was done in the second half of the sixteenth century 
by the Germans Christoph Clavius and Scheubel, by 
the Frenchmen Ramus, Peletier, and Salignac, by 
the Dutchman Menher, and by the Spaniard Nunez. 
It can, therefore, be said that by the end of the six- 
teenth century or the beginning of the seventeenth 
the spirit of the German Coss dominated the Algebra 
of all the European lands, with the single exception 
of Italy. 

The history of the purely arithmetical solution of 
equations of the third and fourth degrees which was 
successfully worked out upon Italian soil demands 
marked attention. Fibonacci (Leonardo of Pisa)* 
made the first advance in this direction in connection 

* Cantor, IL, p. 43 


with the equation x^ + 2^^ 4. iq^ = 20. Although he 
succeeded in solving this only approximately, it fur- 
nished him with the opportunity of proving that the 
value of X cannot be represented by square roots 
alone, even when the latter are chosen in compound 
form, like 

The first complete solution of the equation a^-^-mx^n 
is due to Scipione del Ferro, but it is lost* The 
second discoverer is not Cardan, but Tartaglia. On 
the twelfth of February, 1535, he gave the formula 
for the solution of the equation x^-\-mx = n, which 
has since become so famous under the name of his 
rival. By 1541 Tartaglia was able to solve any equa- 
tion whatsoever of the third degree. In 1539 Cardan 
enticed his opponent Tartaglia to his house in Milan 
and importuned him until the latter finally confided 
his method under the pledge of secrecy. Cardan broke 
his word, publishing Tartaglia's solution in 1545 in 
his Ars magna, although not without some mention of 
the name of the discoverer. Cardan also had the satis 
faction of giving to his contemporaries, in his Ars 
magna, the solution of the biquadratic equation which 
his pupil Ferrari had succeeded in obtaining. Bom- 
belli is to be credited with representing the roots of 
the equation of the third degree in the simplest form, 
in the so-called irreducible case, by means of a trans- 
formation of the irrational quantities. Of the German 

* Hankel, p. 360. 


mathematicians, Rudolff also solved a few equations 
of the third degree, but without explaining the method 
which he followed. Stifel by this time was able to 
give a brief account of the ''cubicoss," that is, the 
theory of equations of the third degree as given in 
Cardan's work. The first complete exposition of the 
Tartaglian solution of equations of the third degree 
comes from the pen of Faulhaber (1604). 

The older cossists* had arranged equations of the 
first, second, third, and fourth degrees (in so far as 
they allow of a solution by means of square roots 
alone) in a table containing twenty-four different 
forms. The peculiar form of these rules, that is, of 
the equations with their solutions, can be seen in the 
following examples taken from Riese : 

**The first rule is when the root [of the equation] 

is equal to a numj^er, or dragma so called. Divide 

by the number of roots ; the result of this division 

must answer the question." (I. e., if ax=^by then 


**The sixteenth rule is when squares equal cubes 

and fourth powers. Divide through by the number 

of fourth powers [the coefficient of :v*], then take half 

the number of cubes and multiply this by itself, add 

this product to the number of squares, extract the 

square root, and from the result take half the number 

of cubes. Then you have the answer.** 

* Treudeia. 


Taking this step by step we have, 

ax^ + ^^ = ^^*> ^* H ^ = — ^, or 

a a 

The twenty-four forms of the older cossists are re- 
duced by Riese to "acht equationes" (eight equa- 
tions, as his combination of German and Latin means), 
but as to the fact that the square root is two-valued 
he is not at all clear. Stifel was the first to let a single 
equation stand for these eight, and he expressly as- 
serts that a quadratic can have only two roots ; this 
he asserts, however, only for the equation x^=^ax — b. 
In order to reduce the equations above mentioned to 
one of Riese*s eight forms, Rudolff availed himself of 
** four precautions (Cautelen)," from which it is clearly 
seen what labor it cost to develop the coss step by 
step. For example, here is his 

*' First precaution. When in equating two num- 
bers, in the one is found a quantity, and in the other 
is found one of the same name, then (considering the 
signs -f- and — ^) must one of these quantities be added 
to or subtracted from the other, one at a time, care 
being had to make up for the defect in the equated 
numbers by subtracting the + and adding the — ." 
(I. e., from f}x^ — Zx^A: = 2x'^'\-hxy we derive 3^:2^ 

The first examples of this period, of equations 
with more than one unknown quantity, are met with 


in Rudolff,* who treats them only incidentally. Here 
also Stifel went decidedly beyond his predecessors. 
Besides the first unknown, Ix, he introduced lA, IB, 
IC, . . .as secundae radices or additional unknowns 
and indicated the new notation made necessary in the 
performance of the fundamental operations, as '^xA 
{=^xy), 1^5 (=j/*), and several others. 

Cardan, over whose name a shadow has been cast 
by his selfishness in his intercourse with Tartaglia, is 
still deserving of credit, particularly for his approxi- 
mate solution of equations of higher degrees by means 
of the regula falsi which he calls regula aurea. Vieta 
went farther in this direction and evolved a method 
of approximating the solution of algebraic equations 
of any degree whatsoever, the method improved by 
Newton and commonly ascribed to him. Reymers and 
Biirgi also contributed to these methods of approxi- 
mation, using the regula falsi. We can therefore say 
that by the beginning of the seventeenth century there 
were practical methods at hand for calculating the 
positive real roots of algebraic equations to any de- 
sired degree of exactness. 

The real theory of algebraic equations is especially 
due to Vieta. He understood (admitting only posi- 
tive roots) the relation of the coeflBcients of equations 
of the second and third degree to their roots, and also 
made the surprising discovery that a certain equation 
of the forty-fifth degree, which had arisen in trig- 

* Cantor, II., p. 392. 


onometric work, possessed twenty-three roots (in this 
enumeration he neglected the negative sine). In Ger- 
man writings there are also found isolated statements 
concerning the analytic theory of equations ; for ex- 
ample, Burgi recognized the connection of a change 
of sign with a root of the equation. However unim- 
portant these first approaches to modern theories may 
appear, they prepared the way for ideas which be- 
came dominant in later times. 




The founding of academies and of royal societies 
characterizes the opening of this period, and is the 
external sign of an increasing activity in the field of 
mathematical sciences. The oldest learned society, 
the Accademia dei Lincei, was organized upon the 
suggestion of a. Roman gentleman, the Duke of Cesi, 
as early as 1603, and numbered, among other famous 
members, Galileo. The Royal Society of London was 
founded in 1660, the Paris Academy in 1C66, and the 
Academy of Berlin in 1700.* 

With the progressive development of pure mathe- 
matics the contrast between arithmetic, which has to 
do with discrete quantities, and algebra, which relates 
rather to continuous quantities, grew more and more 

* Cantor, III., pp. 7, ag. 


marked. Investigations in algebra as well as in the 
theory of numbers attained in the course of time great 

The mighty impulse which Vieta's investigations 
had given influenced particularly the works of Har- 
riot. Building upon Vieta's discoveries, he gave in 
his Ariis analytic ae praxis^ published posthumously in 
the year 1631, a theory of equations, in which the sys- 
tem of notation was also materially improved. The 
signs > and < for ** greater than" and **less than" 
originated with Harriot, and he always wrote x^ for 
XX and jc* for xxx, etc. The sign X ^or "times'* 
is found almost simultaneously in both Harriot and 
Oughtred, though due to the latter; Descartes used 
a period to indicate multiplication, while Leibnitz. in 
1686 indicated multiplication by ^-n and division by v-/, 
although already in the writings of the Arabs the quo- 
tient of a divided by b had appeared in the forms 

a — b, a /by or -7-. The form a:b\s used for the first 


time by Clairaut in a work which was published post- 
humously in the year 1760. Wallis made use in 1655 
of the sign oo to indicate infinity. Descartes made ex- 
tensive use of the the form a* (for positive integral ex- 
ponents). Wallis explained the expressions x"^ and 

X** as indicating the same thing as 1: jr* and vx re- 
spectively ; but Leibnitz and Newton were the first to 
recognize the great importance of, and to suggest, a 
consistent system of notation. 



The powers of a binomial engaged the attention 
of Pascal in his correspondence with Fermat in 1654,* 
which contains the "arithmetic triangle," although, 
in its essential nature at least, it had been suggested 
by Stifel more than a hundred years before. This 
arithmetic triangle is a table of binomial coefficients 
arranged in the following form : 

1 1 





1 2 





1 3 





1 4 





1 5 





1 6 





so that the nth diagonal, extending upwards from left 
to right contains the coefficients of the expansion of 
(tf + dy. Pascal used this table for developing figurate 
numbers and the combinations of a given number of 
elements. Newton generalized the binomial formula 
in 1669, Vandermonde gave an elementary proof in 
1764, and Euler in 1770 in his Anleitung %ur Algebra 
gave a proof for any desired exponent. 

A series of interesting investigations, for the most 
part belonging to the second half of the nineteenth 
century, relates to the nature of number and the ex- 
tension of the number- concept. While among the an- 
cients a "number" meant one of the series of natural 

♦Cantor, II., p. 684. 


numbers only, in the course of time the fundamental 
operations of arithmetic have been extended from 
whole to fractional, from positive to negative, from 
rational and real to irrational and imaginary numbers. * 
For the addition of natural, or integral absolute, 
numbers, which by Newton and Cauchy are often 
termed merely ** numbers, " the associative and com- 
mutative laws hold true, that is, 

Their multiplication obeys the associative, commuta- 
tive, and distributive laws, so that 

To these direct operations correspond, as inverses, 
subtraction and division. The application of these 
operations to all natural numbers necessitates the in- 
troduction of the zero and of negative and fractional 
numbers, thus forming the great domain of rational 
numbers, within which these operations are always 
valid, if we except the one case of division by zero. 

This extension of the number-system showed itself 
in the sixteenth century in the introduction of negative 
quantities. Vieta distinguished affirmative (positive) 
and negative quantities. But Descartes was the first 
to venture, in his geometry, to use the same letter for 
both positive and negative quantities. 

The irrational had been incorporated by Euclid 
into the mathematical system upon a geometric basis, 
this plan being followed for many centuries. Indeed 


it was not until the most modern times* that a purely 
arithmetic theory of irrational numbers was produced 
through the researches of Weierstrass, Dedekind, G. 
Cantor, and Heine. 

Weierstrass proceeds f from the concept of the 
whole number. A numerical quantity consists of a 
series of objects of the same kind ; a number is there- 
fore nothing more than the "combined representation 
of one and one and one, etc. "J By means of subtrac- 
tion and division we arrive af negative and fractional 
numbers. Among the latter there are certain numbers 
which, if referred to one particular system, for exam- 
ple to our decimal system, consist of an infinite num- 
ber of elements, but by transformation can be made 
equal to others arising from the combination of a finite 
number of elements (e. g., 0.1333...=^). These 
numbers are capable of still another interpretation. 
But it can be proved that every number formed from 
an infinite number of elements of a known species, 
and which contains a known finite number of those 
elements, possesses a very definite meaning, whether 
it is capable of actual expression or not. When a 
number of this kind can only be represented by the 
infinite number of its elements, and in no other way, 
it is an irrational number. 

Dedekind§ arranges all positive and negative, in- 

* stole, VorUtungen Uber allgtmeine Aritkmetik, 1885-1886. 

t Kossak, Die EUtnente der Aritkmetik, 1872. 

X Rosier, Die neueren Definitioiuformen der irrationaUn Zaklen, it86. 

S Dedekind, Stetigkeit und irrationale Zaklen^ 1872. 


tegral and fractional numbers, according to their mag- 
nitude, in a system or in a body of numbers {Zahlen 
k5rper)y J?. A given number, a, divides this system 
into the two classes, A\ and A%, each containing in- 
finitely many numbers, so that every number in Ai is 
less than every number in A^, Then a is either the 
greatest number in A\ or the least in A%, These ra- 
tional numbers can be put into a one-to-one corre- 
spondence with the points of a straight line. It is 
then evident that this straight line contains an infinite 
number of other points than those which correspond 
to rational numbers, that is, the system of rational 
numbers does not possess the same continuity as the 
straight line, a continuity possible only by the intro- 
duction of new numbers. According to Dedekind the 
essence of continuity is contained in the following 
axiom : "If all the points of a straight line are divided 
into two classes such that every point of the first class 
lies to the left of every point of the second, then there 
exists one point and only one which effects this divi- 
sion of all points into two classes, this separation of 
the straight line into two parts." With this assump- 
tion it becomes possible to create irrational numbers. 
A rational number, a, produces a Schniti or section 
(y^ijy^j), with respect to A\ and A%, with the charac- 
teristic property that there is in A\ a greatest, or in 
A^ a least number, a. To every one of the infinitely 
many points of the straight line which are not covered 
by rational numbers, or in which the straight line is 



not cut by a rational number, there corresponds one 
and only one section (Ai\Ai), and each one of these 
sections defines one and only one irrational number a. 

In consequence of these distinctions ' ' the system ^ constitutes 
an organized domain of all real numbers of one dimension; by this 
no more is meant to be said than that the following laws govern : * 

I. If a > /CJ, and )3 > 7, then a is also > y ; that is, the number 
P lies between the numbers a, 7. 

II. If a, 7 are two distinct numbers, then there are infinitely 
many distinct numbers P which lie between a and 7. 

III. If a is a definite number, then all numbers of the system 
^ fall into two classes, A^ and A^, each of which contains infinitely 
many distinct numbers; the first class A^^ contains all numbers 
a^ which are <^ a ; the second class A^ contains all numbers a^ 
which are ^ a ; the number a itself can be assigned indifferently 
to either the first or the second class and it is then respectively 
either the greatest number of the first class, or the least of the sec- 
ond. In every case, the separation of the system ^ into the two 
classes A^ and /^j ^^ ^^^^ ^^^^ every number of the first class A^ 
is less than every number of the second class A 2, and we affirm 
that this separation is effected by the number a. 

IV. If the system Id of all real numbers is separated into two 
classes, A^, A^, such that every number a^, of the class A^ is less 
than every number a, of the class A 2, then there exists one and 
only one number a by which this separation is effected (the domain 
H possesses the property of continuity)." 

According to the assertion of J. Tannery f the fundamental 
ideas of Dedekind's theory had already appeared in J. Bertrand's 
text-books of arithmetic and algebra, a statement denied by Dede- 

• Dedekind, SieiigMt und irrationale Zahlen, 1879. 

t Stole, Vorlesungen Uber allgemeine Arithmetik, 1885-1886. 

t Dedekind, Wo* tind und was sollen die Zahlenf 1888. 


G. Cantor and Heine* introduce irrational num- 
bers through the concept of a fundamental series. 
Such a series consists of infinitely many rational num- 
bers,^ 01, Of, ^t, . . . . 0»4^» • • •» ai^d it possesses the 
property that for an assumed positive number c, how- 
ever small, there is an index «, so that for «^«i the 
absolute value of the difference between the term a^ 
and any following term is smaller than c (condition of 
the convergency of the series of the a's). Any two 
fundamental series can be compared with each other 
to determine whether they are equal or which is the 
£reater or the less ; they thus acquire the definiteness 
of a number in the ordinary sense. A number defined 
by a fundamental series is called a "series number." 
A series number is either identical with a rational 
number, or not identical ; in the latter case it defines 
an irrational number. The domain of series numbers 
consists of the totality of all rational and irrational 
numbers, that is to say, of all real numbers, and of 
these only. In this case the domain of real numbers 
can be associated with a straight line, as G. Cantor 
has shown. 

The extension of the number-domain by the addi- 
tion of imaginary quantities is closely connected with 
the solution of equations, especially those of the third 
degree. The Italian algebraists of the sixteenth cen- 
tury called them "impossible numbers." As proper 
solutions of an equation, imaginary quantities first 

* RStler, Die neueren Definilionsformen tier irratianaUn Zahlen, 1886. 


appear m the writings of Albert Girard* (1629). The 
expressions "real" and "imaginary** as characteristic 
terms for the difference in nature of the roots of an 
equation are due to Descartes, f De Moivre and Lam- 
bert introduced imaginary quantities into trigonom- 
etry, the former by means of his famous proposition 
concerning the power (cos <^ + /sin <^)'*, first given in 
its present form by Euler. J 

Gauss§ added to his great fame by explaining the 
nature of imaginary quantities. He brought into gen- 
eral use the sign / for l/ — 1 first suggested by Euler ; || 
he calls a-\-di a complex number with the norm 
cfl+d^. The term ' ' modulus " for the quantity \/a^ + d^ 
comes from Argand (1814), the term "reduced form*' 
for r(cos<^-f-/sin«^), which equals a-\-h\ is due to 
Cauchy, and the name "direction coefficient" for the 
factor cos ^ -f ^ sin ^ first appeared in print in an essay 
of Hankel's (1861), although it was in use somewhat 
earlier. Gauss, to whom in 1799 it seemed simply 
advisable to retain complex numbers,^ by his expla- 
nations in the advertisement to the second treatise on 
biquadratic residues gained for them a triumphant 
introduction into arithmetic operations. 

The way for the geometric representation of com- 
plex quantities was prepared by the observations of 

* Cantor, II., p. 7x8. t Cantor, II., p. 724. t Cantor, III., p. 6B4. 

§ Hankel, Die kontpiexen Zahlerty 1867, p. 71. 

I Beman. "Euler's Use of t to Represent an Imaginary,*' BuU. Amer. 
Math. Soc, March, 1898, p. 274. 



various mathematicians of the seventeenth and eight 
eenth centuries, among them especially WalHs,* who 
in solving geometric problems algebraically became 
aware of the fact that when certain assumptions give 
two real solutions to a problem as points of a straight 
line, other assumptions give two ** impossible" roots 
as the points of a straight line perpendicular to the first 
one. The first satisfactory representation of complex 
quantities in a plane was devised by Caspar Wessel 
in 1797, without attracting the attention it deserved. 
A similar treatment, but wholly independent, was given 
by Argand in 1806. f But his publication was not ap- 
preciated even in France. In the year 1813 there ap- 
peared in Gergonne's Annates by an artillery oflBicer 
Fran9ais in the city of Metz the outlines of a theory 
of imaginary quantities the main ideas of which can 
be traced back to Argand. Although Argand im- 
proved his theory by his later work, yet it did not 
gain recognition until Cauchy entered the lists as its 
champion. It was, however, Gauss who (1831), by 
means of his great reputation, made the representa- 
tion of imaginary quantities in the "Gaussian plane" 
the common property of all mathematicians.]; 

Gauss and Dirichlet introduced general complex 
numbers into arithmetic. The primary investigations 

* Hankel, Bit kompUxen Zahlen, 1867, p. 81. 

t Hankel, Die kompUxen Zahlen, 1867, p. 83. 

X'Pot z. risumi ol the history of the geometric representation of the im- 
aginary, see Beman, "A Chapter in the History of Mathematics," Proc. 
Amer. Assn. Adp. Science^ 1897, pp. 33-50 


of Dirichlet in regard to complex numbers, which, to- 
gether with indications of the proof, are contained in 
the Berichie der Berliner Akademie for 1841, 1842, and 
1846, received material amplifications through Eisen- 
stein, Kummer, and Dedekind. Gauss, in the devel- 
opment of the real theory of biquadratic residues, 
introduced complex numbers of the form a + ^/, and 
Lejeune Dirichlet introduced into the new theory of 
complex numbers the notions of prime numbers, 
congruences, residue- theorems, reciprocity, etc , the 
propositions, however, showing greater complexity 
and variety and offering greater difficulties in the way 
of proof. * Instead of the equation jc* — 1=0, which 
gives as roots the Gaussian units, + 1> — 1> +^» — ^ 
Eisenstein made use of the equation 0^ — li=0 and 
considered the complex numbers a-^-dp (p being a 
complex cube root of unity) the theory resembling that 
of the Gaussian numbers a-^-di, but yet possessing 
certain marked differences. Kummer generalized the 
theory still further, using the equation jp" — 1==0 as 
the basis, so that numbers of the form 

arise where the a's are real integers and the A's aire 
roots of the equation x^ — 1 = 0. Kummer also set 
forth the concept of ideal numbers, that is, of such 
numbers as are factors of prime numbers and possess 
the property that there is always a power of these ideal 
numbers which gives a real number. For example, 

*CftyIe7, Address to the British Association, etc., 1883. 


there exists for the prime number / no rational factor- 
ization so thdit p^=A'B (where A is different from p 
and /*); but in the theory of numbers formed from 
the twenty-third roots of unity there are prime num- 
bers / which satisfy the condition named above. In 
this case / is the product of two ideal numbers, of 
which the third powers are the real numbers A and 
B, so Xh?it p^=A*B. In the later development given 
by Dedekind the units are the roots of any irreducible 
equation with integral numerical coefficients. In the 
case of the equation x^ — jc-|- lr=0, J(l + i\/W), that 
is to say, the p of Eisenstein, is to be regarded as in- 

In tracing out the nature of complex numbers, 
H. Grassmann, Hamilton, and Scheffler have arrived 
at peculiar discoveries. Grassmann, who also mate- 
rially developed the theory of determinants, investi- 
gated in his treatise on directional calculus {Ausdeh- 
nungslehre) the addition and multiplication of complex 
numbers. In like manner, Hamilton originated the 
calculus of quaternions, a method of calculation re- 
garded with especial favor in England and America 
and justified by its relatively simple applicability to 
spherics, to the theory of curvature, and to mechanics. 

The complete double title* of H. Grassmann's 
chief work which appeared in the year 1844, as 
translated, is: ''The Science of Extensive Quantities 
or Directional Calculus {Ausdehnungslehre), A New 

• V. Schlegel, Grassmann, setn Leben und seine IVerke. 



Mathematical Theory, Set Forth and Elucidated by 
Applications. Part First, Containing the Theory of 
Lineal Directional Calculus. The Theory of Lineal 
Directional Calculus, A New Branch of Mathematics, 
Set Forth and Elucidated by Applications to the 
Remaining Branches of Mathematics, as well as to 
Statics, Mechanics, the Theory of Magnetism and 
Crystallography.** The favorable criticisms of this 
wonderful work by Gauss, who discovered that "the 
tendencies of the book partly coincided with the paths 
upon which he had himself been travelling for half a 
century," by Grunert, and by MObius who recognised 
m Grassmann "a congenial spirit with respect to 
mathematics, though not to philosophy," and who 
congratulated Grassmann upon his "excellent work," 
were not able to secure for it a large circle of readers. 
As late as 1853 MObius stated that "Bretschneider 
was the only mathematician in Gotha who had assured 
him that he had read the Ausdehnungslehre through." 
Grassmann received the suggestion for his re- 
searches from geometry, where A^ B, C, being points 
of a straight line, AB^BC=AC.'^ With this he 
combined the propositions which regard the parallelo 
gram as the product of two adjacent sides, thus intro- 
ducing new products for which the ordinary rules of 
multiplication hold so long as there is no permutation 
of factors, this latter case requiring the change of 

* Grassmann, Die Ausdehnungslehre von 1844 oder die lineale Ausdeh- 
nungslehre^ ein neuer Ziveig der Mathentatik. Zweite Aufiae;e, 1878. 


signs. More exhaustive researches led Grassmann to 
regard as the sum of several points their center of 
gravity, as the product of two points the finite line- 
segment between them, as the product of three points 
the area of their triangle, and as the product of four 
points the volume of their pyramid. Through the 
study of the Barycentrischer Calcul of M5bius, Grass- 
mann was led still further. The product of two line- 
segments which form a parallelogram was called the 
''external product" (the factors can be permuted only 
by a change of sign), the product of one line-segment 
and the perpendicular projection of another upon it 
formed the ** internal product" (the factors can here 
be permuted without change of sign). The introduc- 
tion of the exponential quantity led to the enlarge- 
ment of the system, of which Grassmann permitted a 
brief survey to appear in GrunerVs Archiv (1845).* 

Hamiltonf gave for the first time, in a communi- 
cation to the Academy of Dublin in 1844, the values 
/*, y, k^ so characteristic of his theory. The Lectures 
on Quaternions appeared in 1853, the Elements of Qua- 
ternions in 1866. From a fixed point O let a line J be 
drawn to the point P having the rectangular co-ordi- 
nates X, y, z. Now if /, J, k represent fixed coefl&cients 
(unit distances on the axes), then 

* Translated by Beman, Analyst^ 1881, pp. 96, 114. 

t Unvereagt, Theorie der goniometrischen und longimetrischen Quater- 
nionen^ 1876. 

tCayley, A., "On Multiple Algebra," in Quarterly Journal 0/ Mathe- 
matics, 1887. 


is a vector, and this additively joined to the "pure 
quantity" or ** scalar" w produces the quaternion 

Q = w-\-ix -^jy + k%. 

The addition of two quaternions follows from the 
usual formula 


But in the case of multiplication we must place 
fl ==/*^ = ^' = — 1, i=zjk = — kjy J =ki= — ik, 

m • • • • 

k = tj=—jt, 

SO that we obtain 

Q'Q' z=z WW* — XX* — yy — ««' 

+ i{wx* + xw' + ^/ — «y) 
+ j{wy' + yw' + zx' — xz') 
-\- k (wz* -\- zw' + xy' — yx'). 

On this same subject Scheffler published in 1846 
his first work, Ueber die Verhdltnisse der Arithmetik zur 
Geometrie, in 1852 the SHuationscalcul^ and in 1880 the 
Polydimensionalen Grossen. For him * the vector r in 
three dimensions is represented by 

r = a*e^^~^ -e^^^^^y or 
r=a:+j'l/— T + 2|/— ~i-i/^^, or 
r=:x-\'y'i-\-z't'i\ where / = l/ — 1 and ii = \/-^ 1 
are turning factors of an angle of 90° in the plane of xy 
and xz. In Scheffler*s theory the distributive law does 
not always hold true for multiplication, that is to say, 
a{lf'-\- c) is not always equivalent to ad-{- ac. 

Investigations as to the extent of the domain in 

*Unverzagt, Ueber die Grundlagen der Rechnung vtit Quaternionen, 1881. 



which with certain assumptions the laws of the ele- 
mentary operations of arithmetic are valid have led 
to the establishment of a calculus of logic* To this 
class of investigations there belong, besides Grass- 
mann's Formenlehre (1872), notes by Cayley and Ellis, 
and in particular the works of Boole, SchrOder, and 
Charles Peirce. 

A minor portion of the modern theory of numbers 
or higher arithmetic, which concerns the theories of 
congruences and of forms, is made up of continued 
fractions. The algorism leading to the formation of 
such fractions, which is also used in calculating the 
greatest common measure of two numbers, reaches 
back to the time of Euclid. The combination of the 
partial quotients in a continued fraction originated 
with Cataldi,f who in the year 1613 approximated the 
value of square roots by this method, but failed to 
examine closely the properties of the ne*w fractions. 

Daniel Schwenter was the first to make any ma- 
terial contribution (1625) towards determining the 
convergents of continued fractions. He devoted his 
attention to the reduction of fractions involving large 
numbers, and determined the rules now in use for cal- 
culating the successive convergents. Huygens and 
Wallis also labored in this field, the latter discover- 
ing the general rule, together with a demonstration, 
which combines the terms of the convergents 


* SchrOder, Der Of>erationskreis des Logikcalculs^ 1877 
t Cantor, II., p. 693. 


in the following manner : 

Pn __ ^nPn-l + ^nPn^ 
^n ^n^K-l+^n^,^' 

The theory of continued fractions received its greatest 
development in the eighteenth century with Euler,* 
who introduced the name fracHo continua (the Ger- 
man term Kettenbruch has been used only since the 
beginning of the nineteenth century). He devoted 
his attention chiefly to the reduction of continued 
fractions to the form of infinite products and series, 
and doubtless in this way was led to the attempt to 
give the convergents independent form, that is to dis- 
cover a general law by means of which it would be 
possible to calculate any required convergent without 
first obtaining the preceding ones. Although Euler 
did not succeed in discovering such a law, he created 
an algorism of some value. This, however, did not 
bring him essentially nearer the goal because, in spite 
of the example of Cramer, he neglected to make use 
of determinants and thus to identify himself the more 
closely with the pure theory of combinations. From 
this latter point of view the problem was attacked by 
Hindenburg and his pupils Burckhardt and Rothe. 
Still, those who proceed from the theory of combina- 
tions alone know continued fractions only from one 
side ; the method of independent presentation allows 

* Cantor, III., p. 670. 

ALGEBRA. 1 33 

the calculation of the desired convergent from both 
sides, forward as well as backward, to the practical 
value of which Dirichlet has testified. 

Only in modern times has the calculus of determi- 
nants been employed in this field, together with a 
combinatory symbol, and the first impulse in this di- 
rection dates from the Danish mathematician Ramus 
(1855). Similar investigations were begun, however, 
by Heine, M5bius, and S. Gunther, leading to the 
formation of ** continued fractional determinants.'' 
The irrationality of certain infinite continued frac 
tions* had been investigated before this by Legendre, 
who, like Gauss, gave the quotient of two power se- 
ries in the form of a continued fraction. By means of 
the application of continued fractions it can be shown 
that the quantities e' (for rational values of *), ^, ir, 
and «* cannot be rational (Lambert, Legendre, Stern). 
It was not until very recent times that the transcen- 
dental nature of e was established by Hermite, and 
that of IT by F. Lindemann. 

In the theory of numbers strictly speaking, quite 
difficult problems concerning the properties of num- 
bers were solved by the first exponents of that study, 
Euclid and Diophantus. Any considerable advance 
was impossible, however, as long as investigations had 
to be conducted f without an adequate numerical nota- 
tion, and almost exclusively with the aid of an algebra 

♦ Treutlein. 

t Legendre, Tkiorie ties nombres, ist ed. 1798, 3rd ed. 1830. 


just developing under the guise of geometry. Until 
the time of Vieta and Bachet there is no essential ad- 
vance to be noted in the theory of numbers. The 
former solved many problems in this field, and the 
latter gave in his work Problemes plaisanis ei ddlectables 
a satisfactory treatment of indeterminate equations 
of the first degree. Still later the first stones for the 
foundation of a theory of numbers were laid by Fer- 
mat, who had carefully studied Diophantus and into 
whose works as elaborated by Bachet he incorporated 
valuable additional propositions. The great mass of 
propositions which can be traced back to him he gave 
for the most part without demonstration, as for ex- 
ample the following statement : 

"Every prime number of the form 4«-|-l is the 
sum of two squares; a prime number of the form 

^n-\-\ has at the same time the three forms ^' -f ^*> 
^ + 22;', y^ — 2jb' ; every prime number of the form 
8«+ 3 appears as y^ + 22*, every one of the form %n-\-^ 
appears as y^ — 22*." Further, **Any number gan be 
formed by the addition of three cubes, of four squares, 
of five fifth powers, etc." 

Fermat proved that the area of a Pythagorean 
right-angled triangle, for example a triangle with the 
sides 3, 4, and 5, cannot be a square. He was also 
the first to obtain the solution of the equation ax^ -f 
1 =j'*, where a is not a square ; at all events, he 
brought this problem to the attention of English 
mathematicians, among whom Lord Brouncker dis- 


covered a solution which found its way into the 
works of Wallis. Many of Fermat's theorems belong 
to '*the finest propositions of higher mathematics,"* 
and possess the peculiarity that they can easily be 
discovered by induction, but that their demonstrations 
are extremely difficult and yield only to the most 
searching investigation. It is just this which imparts 
to higher arithmetic that magic charm which made it 
a favorite with the early geometers, not to speak of 
its inexhaustible treasure-house in which it far ex- 
ceeds all other branches of pure mathematics. 

After Fermat, Euler was the first again to attempt 
any serious investigations in the theory of numbers. 
To him we owe, among other things, the first scien- 
tific solution of the chess board problem, which re- 
quires that the knight, starting from a certain square, 
shall in turn occupy all sixty-four squares, and the 
further proposition that the sum of four squares mul- 
tiplied into another similar sum also gives the sum of 
four squares. He also discovered demonstrations of 
various propositions of Fermat, as well as the general 
solution of indeterminate equations of the second de- 
gree with two unknowns on the hypothesis that a spe- 
cial solution is known, and he treated a large number 
of other indeterminate equations, for which he dis- 
covered numerous ingenious solutions. 

Euler (as well as Krafft) also occupied himself 

*Gaa88, fVerAt, II., p. 152. 


With amicable numbers.* These numbers, which are 
mentioned by lamblichus as being known to the 
Pythagoreans, and which are mentioned by the Arab 
Tabit ibn Kurra, suggested to Descartes the discovery 
of a law of formation, which is given again by Van 
Schooten. Euler made additions to this law and de- 
duced from it the proposition that two amicable num 
bers must possess the same number of prime factors. 
The formation of amicable numbers depends either 
upon the solution of the equation xy-\- ax-\-fy-\- c^=:0, 
or upon the factoring of the quadratic form aa^ -\- bxy 

Following 'Euler, Lagrange was able to publish 
many interesting results in the theory of numbers. 
He showed that any number can be represented as 
the sum of four or less squares, and that a real root 
of an algebraic equation of any degree can be con- 
verted into a continued fraction. He was also the 
first to prove that the equation x^- — Ay^=z\ is always 
soluble in integers, and he discovered a general method 
for the derivation of propositions concerning prime 

Now the development of the theory of numbers 
bounds forward in two mighty leaps to Legendre and 
Gauss. The valuable treatise of the former, Essai sur 
la thiorie des nombres, which appeared but a few years 
before Gauss's Disquisitiones arithmeticaey contains an 
epitome of all results that had been published up to 

*Seelhoff, ** Befreundete Zahlen," Hoppe Arch,, Bd. 70. 


that time, besides certain original investigations, the 
most brilliant being the law of quadratic reciprocity, 
or, as Gauss called it, the Theorema fundameniale in 
doetrina de residuis quadratis. This law gives a rela- 
tionship between two odd and unequal prime numbers 
and can be enunciated in the following words : 

**Let ( — I be the remainder which is left after divid- 


ing m ^ by «, and let ^ be the remainder left after 

dividing n ^ by m. These remainders are always 
-f- 1 or — 1. Whatever then the prime numbers m 

and n may be, we always obtain ( — ) = ( — ) iii case the 
numbers are not both of the form 4^ + 3. But if both 

are of the form 4:x -f 3, then we have ( — ) = — ( ~ )• " 
These two cases are contained in the formula 

Bachet having exhausted the theory of the indetermi- 
nate equation of the first degree with two unknowns^, 
an equation which in Gauss's notation appears in the 

form x'^a (mod ^), identical with -j- =y-\-—, mathe- 
maticians began the study of the congruence ^^^w 
(mod «). Fermat was aware of a few special cases of 
the complete solution ; he knew under what conditions 
± 1, 2, ± 3, 5 are quadratic residues or non-residues 
of the odd prime number »/.* For the cases — 1 and 

"= Baumgart, "Ueber das quadratische Reciprocit&tsgesets," in Schl9- 
milch' s Zeitschri/if Bd. 30, HI. Abt. 


±3 the demonstrations originate with Euler, for ±2 
and ± 5 with Lagrange. It was Euler, too, who gave 
the propositions which embrace the law of quadratic 
reciprocity in the most general terms, although he 
did not offer a complete demonstration of it. The 
famous demonstration of Legendre (in Essai sur la 
thiorie des nombres, 1798) is also, as yet, incomplete. 
In the year 1796 Gauss submitted, without knowing 
of Euler's work, the first unquestionable demonstra- 
tion — a demonstration which possesses at the same 
time the peculiarity that it embraces the principles 
which were used lateif. In the course of time Gauss 
adduced no less than eight proofs for this important 
law, of which the sixth (chronologically the last) was 
simplified almost simultaneously by Cauchy, Jacobi, 
and Eisenstein. Eisenstein demonstrated in partic- 
ular that the quadratic, the cubic and the biquadratic 
laws are all derived from a common source. In the 
year 1861 Kummer worked out with the aid of the 
theory of forms two demonstrations for the law of 
quadratic reciprocity, which were capable of gene- 
ralization for the «th-power residue. Up to 1890 
twenty-five distinct demonstrations of the law of 
quadratic reciprocity had been published ; they make 
use of induction and reduction, of the partition of the 
perigon, of the theory of functions, and of the theory 
of forms. In addition to the eight demonstrations by 
Gauss which have already been mentioned, there are 
four by Eisenstein, two by Kummer, and one each 

ALGEBRA. 1 39 

by Jacobi, Cauchy, Liouville, Lebesgue, Genocchi, 
Stern, Zeller, Kronecker, Bouniakowsky, Schering, 
Petersen, Voigt, Busche, and Pepin. 

However much is due to the co-operation of math- 
ematicians of different periods, yet to Gauss unques- 
tionably belongs the merit of having contributed in 
his Disquisitiones arithmeticae of 1801 the most impor- 
tant part of the elementary development of the theory 
of numbers. Later investigations in this branch have 
their root in the soil which Gauss prepared. Of such 
investigations, which were not pursued until after the 
introduction of the theory of elliptic transcendents, 
may be mentioned the propositions of Jacobi in regard 
to the number of decompositions of a number into 
two, four, six, and eight squares,* as well as the in- 
vestigations of Dirichlet in regard to the equation 

His work in the theory of numbers was Dirichlet's 
favorite pursuit. f He was the first to deliver lectures 
on the theory of numbers in a German university and 
was able to boast of having made the Disquisitiones 
arithmeticae of Gauss transparent and intelligible — a 
task in which a Legendre, according to his own 
avowal, was unsuccessful. 

Dirichlet 's earliest treatise, Mimoire sur I ^impossi- 
bility de quelques Equations inddterminis du cinquieme 
degri (submitted to the French Academy in 1825), 

* Dirichlet, " GedSchtnisrede auf Jacobi," Crelle's Journal, Bd. 52. 

\ Kummer, " Gedachtnisrede auf Lejeane-Dirichlet," in Berl. Abh, i86ob 


deals with the proposition, stated by Fermat without 
demonstration, that ** the sum of two powers having 
the same exponent can never be equal to a power of 
the same exponent, when these powers are of a degree 
higher than the second." Euler and Legendre had 
proved this proposition for the third and fourth pow- 
ers ; Dirichlet discusses the sum of two fifth powers 
and proves that for integral numbers x^ +y^ cannot 
be equal to dz^. The importance of this work lies in 
its intimate relationship to the theory of forms of 
higher degree. Dirichlet's further contributions in the 
field of the theory of numbers contain elegant demon- 
strations of certain propositions of Gauss in regard 
to biquadratic residues and the law of reciprocity, 
which were published in 1825 in the Gottingen Ge- 
lehrte Anzeigen^ as well as with the determination of 
the class-number of the quadratic form for any given 
determinant. His ''applications of analysis to the 
theory of numbers are as noteworthy in their way as 
Descartes's applications of analysis to geometry. They 
would also, like the analytic geometry, be recognized 
as a new mathematical discipline if they had been ex- 
tended not to certain portions only of the theory of 
number, but to all its problems uniformly.* 

The numerous investigations into the properties 
and laws of numbers had led in the seventeenth cen- 
turyt to the study of numbers in regard to their divis- 

♦ Kummer, " Gedftchtnisrede auf Lejeane-Dirichlet." Berl, Abh. i860. 
tSeelhoff, "Geschichte der Faktorentafeln," in Hoppe Arch., Bd. 70. 


ors. For almost two thousand years Eratosthenes*s 
** sieve" remained the only method of determining 
prime numbers. In the year 1657 Franz van Schooten 
published a table of prime numbers up to ten thou- 
sand. Eleven years later Pell constructed a table of 
the least prime factors (with the exception of 2 and 5) 
of all numbers up to 100 000. In Germany these 
tables remained almost unknown, and in the year 
1728 Poetius published independently a table of fac- 
tors for numbers up to 100 000, an example which 
was repeatedly imitated. Kriiger's table of 1746 in- 
cludes numbers up to 100 000 ; that of Lambert of 
1770, which is the first to show the arrangement 
used in more modern tables, includes numbers up to 
102 000. Of the six tables which were prepared be- 
tween the years 1770 and 1811 that of Felkel is inter- 
esting because of its singular fate ; its publication by 
the Kaiserlich konigliches Aerarium in Vienna was 
completed as far as 408 000 ; the remainder of the 
manuscript was then withheld and the portion already 
printed was used for manufacturing cartridges for the 
last Turkish war of the eighteenth century. In the 
year 1817 there appeared in Paris Burckhardt's Table 
des diviseurs pour tous les notnbres du i^y <2', j' million. 
Between 1840 and 1850 Crelle communicated to the 
Berlin Academy tables of factors for the fourth, fifth, 
and sixth million, which, however, were not pub- 
lished. Dase, who is known for his arithmetic gen- 
ius, was to make the calculations for the seventh to 


the tenth million, having been designated for that 
work by Gauss, but he died in 1861 before its com- 
pletion. Since 1877 the British Association has been 
having these tables continued by Glaisher with the 
assistance of two computers. The publication of 
tables of factors for the fourth million was completed 
in 1879.' 

In the year 1856 K. G. Reuschle published his 
tables for use in the theory of numbers, having been 
encouraged to undertake the work by his correspond- 
ence with Jacobi. They contain the resolution of 
numbers of the form 10* — 1 into prime factors, up to 
«=:242, and numerous similar results for numbers of 
the form a** — 1, and a table of the resolution of prime 
numbers / = 6« + 1 into the forms . 

/ = ^« + 3^ and 4/=C> + 27J/», 

as they occur in the treatment of cubic residues and 
in the partition of the perigon. 

Of greatest importance for the advance of the sci- 
ence of algebra Sis well as that of geometry was the 
development of the theories of symmetric functions, 
of elimination, and of invariants of algebraic forms, 
as they were perfected through the application of pro- 
jective geometry to the theory of equations.* 

The first formulas for calculating symmetric func- 
tions (sums of powers) of the roots of an algebraic 
equation in terms of its coefficients are due to Girard. 

*A. Brill, Antrittsrede in Tubingen, 1884. Manuscript. 

ALGEBRA. 1 43 

Waring also worked in this field (1770) and developed 
a theorem, which Gauss independently discovered 
(1816), by means of which any symmetric function 
may be expressed in terms of the elementary sym- 
metric functions. This is accomplished directly by a 
method devised by Cay ley and Sylvester, through laws 
due to the former in regard to the weight of sym 
metric functions. The oldest tables of symmetric 
functions (extending to the tenth degree) were pub 
lished by Meyer- Hirsch in his collection of problems 
(1809). The calculation of these functions, which was 
very tedious, was essentially simplified by Cay ley and 

The resultant of two equations with one unknown, 
or, what is the same, of two forms with two homo- 
geneous variables, was given by Euler (1748) and by 
B^zout (1764). To both belongs the merit of having 
reduced the determination of the resultant to the so- 
lution of a system of linear equations.* B^zout intro- 
duced the name *' resultant" (De Morgan suggested 
**eliminant") and determiped the degree of this func- 
tion. Lagrange and Poisson also investigated ques- 
tions of elimination ; the former stated the condition 
for common multiple-roots; the latter furnished a 
method of forming symmetric functions of the com- 
mon values of the roots of a system of equations. The 
further advancement of the theory of elimination was 
made by Jacobi, Hesse, Sylvester, Cayley, Cauchy, 

♦ Salmon, Higher Algtbra. 


Brioschi, and Gordan. Jacobi's memoir,* which rep- 
resented the resultant as a determinant, threw light 
at the same time on the aggregate of coefficients be- 
longing to the resultant and on the equations in which 
the resultant and its product by another partially ar- 
bitrary function are represented as functions of the 
two given forms. This notion of Jacobi gave Hesse 
the impulse to pursue numerous important investiga- 
tions, especially on the resultant of two equations, 
which he again developed in 1843 after Sylvester's 
dialytic method (1840); then in 1844, *'on the elimi- 
nation of the variables from three algebraic equations 
with two variables"; and shortly after **on the points 
of inflexion of plane curves." Hesse placed the main 
value of these investigations, not in the form of the 
final equation, but in the insight into the composition 
of the same from known functions. Thus he came 
upon the functional determinant of three quadratic 
prime forms, and further upon the determinant of the 
second partial differential coefficients of the cubic 
form, and upon its Hessian determinant, whose geo- 
metric interpretation furnished the interesting result 
that in the general case the points of inflexion of a 
plane curve of the nth order are given by its complete 
intersection with a curve of order 3(« — 2). This re- 
sult was previously known for curves of the third 
order, having been discovered by Pliicker. To Hesse 
is further due the first important example of the re- 

♦Noether, Sckldmilch*s Zeitschrift^ Bd. 20. 


moval of factors from resultants, in so far as these 
factors are foreign to the real problem to be solved. 
Hesse, always extending the theory of elimination, 
in 1849 succeeded in producing, free from all super- 
fluous factors, the long-sought equation of the 14th 
degree upon which the double tangents of a curve of 
the 4th order depend. 

The method of elimination used by Hesse* in 1843 
is the dialytic method published by Sylvester in 1840 ; 
it gives the resultant of two functions of the mth and 
nth orders as a determinant, in which the coefficients 
of the first enter into n rows, and those of the second 
into Pi rows. It was Sylvester also, who in 1851 in- 
troduced the name ''discriminant " for the function 
which expresses the condition for the existence of 
two equal roots of an algebraic equation ; up to this 
time, it was customary, after the example of Gauss, 
to say ** determinant of the function." 

The notion of invariance, so important for all 
branches of mathematics to-day, dates back in its 
beginnings to Lagrangef, who in 1773 remarked 
that the discriminant of the quadratic form aa^ -{- 
Uxy-\-cy^ remains unaltered by the substitution of 
x^\y for X. This unchangeability of 'the discrim- 
inant by linear transformation, for binar}^ and ternary 
quadratic forms, was completely proved by Gauss 
(1801) ; but that the discriminant in general and in 
every case remains invariant by linear transformation, 

♦ Matthiessen, p. 99. t Salmon, Higher Algebra, 


G. Boole (1841) recognized and first demonstrated 
In 1845, Cayley, adding to the treatment of Boole, 
found that there are still other functions which possess 
invariant properties in linear transformation, showed 
how to determine such functions and named them 
"hyperdeterminants." This discovery of Cayley de- 
veloped rapidly into the important theory of invari- 
ants, particularly through the writings of Cayley, 
Aronhold, Boole, Sylvester, Hermite, and Brioschi, 
and then through those of Clebsch, Gordan, and 
others. After the appearance of Cayley's first paper, 
Aronhold made an important contribution by deter- 
mining the invariants S and T'of a ternary form, and 
by developing their relation to the discriminant of 
the same form. From 1851 on, there appeared a se- 
ries of important articles by Cayley and Sylvester. 
The latter created in these a large part of the termin 
ology of to-day, especially the name "invariant" 
(1851). In the year 1854, Hermite discovered his law 
of reciprocity, which states that to every covariant or 
invariant of degree p and order r of a form of the mth 
order, corresponds also a covariant or invariant of 
degree m and of order r of a form of the pth order. 
Clebsch and Gordan used the abbreviation d^, intro- 
duced for binary forms by Aronhold, in their funda- 
mental developments, e. g., in the systematic ex- 
tension of the process of transvection in forming 
invariants and covariants, already known to Cayley 
in his preliminary investigations, in the folding- pro- 


cess of forming elementary covariants, and in the for- 
mation of simultaneous invariants and covariants, in 
particular the combinants. Gordan's theorem on the 
finiteness of the form- system constitutes the most im- 
portant recent advance in this theory ; this theorem 
states that there is only a finite number of invariants 
and covariants of a binary form or of a system of such 
forms. Gordan has also given a method for the for- 
mation of the complete form-system, and has carried 
out the same for the case of binary forms of the fifth 
and sixth orders. Hilbert (1890) showed the finite- 
ness of the complete systems for forms of n variables.* 

To refer in a word to the great significance of the theory of 
invariants for other branches of mathematics, let it sufi^ce to 
mention that the theory of binary forms has been transferred by 
Clebsch to that of ternary forms (in particular for equations in 
line co-ordinates) ; that the form of the third order finds its repre- 
sentation in a space-curve of the third order, while binary forms 
of the fourth order play a great part in the theory of plane curves 
of the third order, and assist in the solution of the equation of 
the fourth degree as well as in the transformation of the elliptic 
integral of the first class into Hermite's normal form ; finally that 
combinants can be effectively introduced in the transformation of 
equations of the fifth and sixth degrees. The results of investiga- 
tions by Clebsch, Weierstrass, Klein, Bianchi, and Burckhardt, 
have shown the great significance of the theory of invariants for 
the theory of the hyperelliptic and Abelian functions. This theory 
has been further used by Christoffel and Lipschitz in the represen- 
tation of the line-element, by Sylvester, Halphen, and Lie in the 
case of reciprocants or differential invariants in the theory of dif- 

* Meyer, W. F., " Bericht iiber den gegenwartigen Stand der Invarianten- 
theorie." Jahresbericht der deutschen Mathemaiiker-Vereinigungy Bd. I. 


ferential equations, and by Beltrami in his dififerential parameter 
in the theory of curvature of surfaces. Irrational invariants also 
have been proposed in various articles by Hilbert. 

The theory of probabilities assumed form unde^: 
the hands of Pascal and Fermat.* In the year 1654, 
a gambler, the Chevalier de M^r6, had addressed two 
inquiries to Pascal as follows : ** In how many throws 
with dice can one hope to throw a double six," and 
' ' In what ratio should the stakes be divided if the 
game is broken up at a given moment?" These two 
questions, whose solution was for Pascal very easy, 
were the occasion of his la3ring the foundation of a 
new science which was named by him "G6om6trie du 
hasard." At Pascal's invitation, Fermat also turned 
his attention to such questions, using the theory of 
combinations. Huygens soon followed the example 
of the two French mathematicians, and wrote in 1656f 
a small treatise on games of chance. The first to 
apply the new theory to economic sciences was the 
"grand pensioner" Jean de Witt, the celebrated pupil 
of Descartes. He made a report in 1671 on the man- 
ner of determining the rate of annuities on the basis 
of a table of mortality. Hudde also published in- 
vestigations on the same subject "Calculation of 
chances" (Rechnurtg uber den Zufall^ received compre- 
hensive treatment at the hand of Jacob Bernoulli in 
his Ars conjectandi (1713), printed eight years after the 
death of the author, a book which remained forgotten 

* Cantor, II., p. 68S. t Cantor. II., p. 69a. 


until Condorcet called attention to it. Since Ber- 
noulli, there has scarcely been a distinguished alge- 
braist who has not found time for some work in the 
theory of probabilities. 

To the method of least squares Legendre gave the 
name in a paper on this subject which appeared in 
1805.* The first publication by Gauss on the same 
subject appeared in 1809, although he was in posses- 
sion of the method as early as 1795. The honor is 
therefore due to Gauss for the reason that he first set 
forth the method in its present form and turned it to 
practical account on a large scale. The apparent in- 
spiration for this investigation was the discovery of 
the first planetoid Ceres on the first of January, 1801, 
by Piazzi. Gauss calculated by new methods the 
orbit of this heavenly body so accurately that the 
same planetoid could be again found towards the end 
of the year 1801 near the position given by him. The 
investigations connected with this calculation ap- 
peared in 1809 as Theoria motus corporum coelestium, 
etc. The work contained the determination of the 
position of a heavenly body for any given time by 
means of the known orbit, besides the solution of the 
difiicult problem to find the orbit from three observa- 
tions. In order to make the orbit thus determined 
agree as closely as possible with that of a greater 
number of observations, Gauss applied the process 

*Merriinan, M., "List of Writings relating to the Method of Least 
Squares." Trans. Conn. Acad., Vol. IV. 



discovered by him in 1795. The object of this was 
'<80 to combine observations which serve the purpose 
of determining unknown quantities, that the unavoid- 
able errors of observation affect as little as possible 
the values of the numbers sought." For this purpose 
Gauss gave the following rule*: ''Attribute to each 
error a moment depending upon its value, multiply 
the moment of each possible error by its probability 
and add the products. The error whose moment is 
equal to this sum will have to be designated as the 
mean." As the simplest arbitrary function of the 
error which shall be the moment of the latter, Gauss 
chose the square. Laplace published in the year 1812 
a detailed proof of the correctness of Gauss's method. 
Elementary presentations of the theory of combi- 
nations are found in the sixteenth century, e. g., by 
Cardan, but the first great work is due to Pascal. In 
this he uses his arithmetic triangle, in order to de- 
termine the number of combinations of m elements of 
the nth class. Leibnitz and Jacob Bernoulli produced 
much new material by their investigations. Towards 
the end of the eighteenth century, the field was cul- 
tivated by a number of German scholars, and there 
arose under the leadership of Hindenburg the ''com- 
binatory school, "f whose followers added to the de- 
velopment of the binomial theorem. Superior to them 
all in systematic proof is Hindenburg, who separated 

*Gerhardt, Geschichte der Matkematik in Deutsch/anif, 1877. 
t Gerhardt, Geschichte der Mathemaiik in Deutschland^ 1877. 

ALGEBRA. 15 1 

polynomials into a first class of the form a^-^-f r-f 
//+ • • • and into a second, a-\-bX'\' csfl + dofi -f • • • • 
He perfected what was already known, and gave the 
lacking proofs to a number of theorems, thus earning 
the title of "founder of the theory of combinatory 

The combinatory school, which inclnded Bschenbach, Rothe. 
and especially Pfaff, in addition to its distioguished founder, pro- 
duced a varied literature, and commanded respect because of its 
elegant formal results. But, in its aims, it stood so far outside the 
domain of the new and fruitful theories cultivated especially by 
such French mathematicians as Lagrange and Laplace, that it re- 
mained without influence in the further development of mathemat- 
ics, at least at the beginning of the nineteenth century. 

In the domain of infinite series,* many cases which 
reduce for the most part to geometric series, were 
treated by Euclid, and to a greater degree by Apol- 
lonius. The Middle Ages added nothing essential, 
and it remained for more recent generations to make 
important contributions to this branch of mathemat- 
ical knowledge. Saint- Vincent and Mercator devel- 
oped independently the series for log(l + ^), Gregory 
those for tan~^jc, sinjc, cosjt, secjr, cosecjc. In the 
writings of the latter are also found, in the treatment 
of infinite series, the expressions "convergent" and 
"divergent." Leibnitz was led to infinite series, 
through consideration of finite arithmetic series. He 
realized at the same time the necessity of examining 

* Rei£E, R., Gesckichte der unendlichen Reihen, Tiibingen, 1889. 


more closely into the convergence and divergence of 
series. This necessity was also felt by Newton, who 
used infinite series in a manner similar to that of 
Apollonius in the solution of algebraic and geometric 
problems, especially in the determination of areas, 
and consequently as equivalent to integration. 

The new ideas introduced by Leibnitz were further 
developed by Jacob and John Bernoulli. The former 
found the sums of series with constant terms, the lat- 
ter gave a general rule for the development of a func- 
tion into an infinite series. At this time there were 
no exact criteria for convergence, except those sug- 
gested by Leibnitz for alternating series. 

During the years immediately following, essential 
advances in the formal treatment of infinite series 
were made. De Moivre wrote on recurrent series and 
exhausted almost completely their essential proper- 
ties. Taylor's and Maclaurin's closely related series 
appeared, Maclaurin developing an imperfect proof of 
Taylor's theorem, giving numerous applications of it, 
and stating new formulas of summation. Euler dis- 
played the greatest skill in the handling of infinite 
series, but troubled himself little about convergence 
and divergence. He deduced the exponential from 
the binomial series, and was the first to develop ra- 
tional functions into series of sines and cosines of 
integral multiple arguments.* In this manner he 
defined the coefficients of a trigonometric series by 

* Reiff, Gesckickte der unendlichen Reihen, 1889, pp. 105, 127. 


definite integrals without applying these important 
formulas to the development of arbitrary functions 
into trigonometric series. This was first accomplished 
by Fourier (1822), whose investigations were com- 
pleted by Riemann and Cauchy. The investigation 
was brought to a temporary termination by Dirichlet 
(1829), in so far as by rigid methods he gave it a sci- 
entific foundation and introduced general and com- 
plex investigations on the convergence of series.* 
From Laplace date the developments into series of 
two variables, especially into recurrent series. Le- 
gendre furnished a valuable extension of the theory 
of series by the introduction of spherical functions. 

With Gauss begin more exact methods of treat- 
ment in this as in nearly all branches of mathematics, 
the establishment of the simplest criteria of conver- 
gence, the investigation of the remainder, and the 
continuation of series beyond the region of conver- 
gence. The introduction to this was the celebrated 
series of Gauss : 

^^ l-y^^ l-2-y(y-fl) ^"t* • • • > 

which Euler had already handled but whose great 
value he had not appreciated, f The generally ac- 
cepted naming of this series as "hypergeometric" is 
due to J. F. Pfaff, who proposed it for the general 
series in which the quotient of any term divided by the 

* Kummer, " Gedilchtnissrede auf Lejenne-Dirichlet." Berliner Ahhand- 
iungen, i860. 

t Reiff, GttchichU der unendlichen Reihen, 1889, p. 161. 



preceding is a function of the index. Euler, follow- 
ing WalliSy used the same name for the series in which 
that quotient is an integral linear function of the 
index.* Gauss, probably influenced by astronomical 
applications, stated that his series, by assuming cer- 
tain special values of a, fi, y, could take the place 
of nearly all the series then kno¥m ; he also investi- 
gated the essential properties of the function repre- 
sented by this series and gave for series in general an 
important criterion of convergence. We are indebted 
to Abel (1826) for important investigations on the con- 
tinuity of series. 

The idea of uniform convergence arose from the 
study of the behavior of series in the neighborhood of 
their discontinuities, and was expressed almost simul- 
taneously by Stokes and Seidel (1847-1848). The 
latter calls a series uniformly convergent when it rep- 
resents a discontinuous function of a quantity x, the 
separate terms of which are continuous, but in the 
vicinity of the discontinuities is of such a nature that 
values of x exist for which the series converges as 
slowly as desired, f 

On account of the lack of immediate appreciation 
of Gauss's memoir of 1812, the period of the discovery 
of effective criteria of convergence and divergence]! 
may be said to begin with Cauchy (1821). His meth- 

* Riemann, ff^trkty p. 78. 

t ReifE, GeschichU der unendlichen Rtihtn^ 1889, p. 907. 

tPringsheim, *'Allgemeine Theorie der Divergenz und Konvergens von 
Reihen mit positiven Gliedern," Math. Annalen^ XXXV. 


ods of investigation, as well as the theorems on in- 
finite series with positive terms published between 
1832 and 1851 by Raabe, Duhamel, De Morgan, Ber- 
trand, Bonnet, and Paucker, set forth special criteria, 
for they compare generally the nth term with particu- 
lar functions of the form a", «*, «(log«)* and others. 
Criteria of essentially more general nature were first 
discovered by Kummer (1835), and were generalized 
by Dini (1867). Dini's researches remained for a 
time, at least in Germany, completely unknown. Six 
years later Paul du Bois-Reymond, starting with the 
same fundamental ideas as Dini, discovered anew the 
chief results of the Italian mathematician, worked 
them out more thoroughly and enlarged them essen- 
tially to a system of convergence and divergence cri- 
teria of the first and second kind, according as the 
general term of the series a^ or the quotient a^, : a^ is 
the basis of investigation. Du Bois- Key mend's re- 
sults were completed and in part verified somewhat 
later by A. Pringsheim. 

After the solution of the algebraic equations of the 
third and fourth degrees was accomplished, work on 
the structure of the system of algebraic equations in 
general could be undertaken. Tartaglia, Cardan, and 
Ferrari laid the keystone of the bridge which led from 
the solution of equations of the second degree to the 
complete solution of equations of the third and fourth 
degrees. But centuries elapsed before an Abel threw 
a flood of light upon the solution of higher equations. 


Vieta had found a means of solving equations allied 
to evolution, and this was further developed by Harriot 
and Oughtred, but without making the process less 
tiresome.* Harriot's name is connected with another 
theorem which contains the law of formation of the 
coefficients of an algebraic equation from its roots, 
although the theorem was first stated in full by Des- 
cartes and proved general by Gauss. 

Since there was lacking a sure method of deter- 
mining the roots of equations of higher degree, the 
attempt was made to include these roots within as 
narrow limits as possible. De Beaune and Van 
Schooten tried to do this, but the first usable methods 
date from Maclaurin {Algebra^ published posthum- 
ously in 1748) and Newton (1722) who fixed the real 
roots of an algebraic equation between given limits. 
In order to effect the general solution of an algebraic 
equation, the effort was made either to represent the 
given equation as the product of several equations of 
lower degree, a method further developed by Hudde, 
or to reduce, through extraction of the square root, 
an equation of even degree to one whose degree is 
half that of the given equation ; this method was used 
by Newton, but he accomplished little in this direc- 

Leibnitz had exerted himself as strenuously as 
Newton to make advances in the theory of algebraic 
equations. In one of his letters he states that he has 

♦ Montucla, ffisiotre des Mathlmatiques, 1799-1802. 


ALGEBRA. 1 57 

been engaged for a long time in attempting to find 
the irrational roots of an equation of any degree, by 
eliminating the intermediate terms and reducing it to 
the form x^^^A^ and that he was persuaded that in 
this manner the complete solution of the general equa- 
tion of the «th degree could be effected. This method 
of transforn^ation of the general equation dates back 
to Tschirnhausen and is found as "Nova methodus 
etc." in the Leipziger Acta eruditorum of the year 1683. 
In the equation 

X^ + Ax^^ -{- Bx'^ + +Mx + J\r=0 

Tschirnhausen places 

y=a + Px + yx^ + ...+lix^^; 

the elimination of x from these two equations gives 
likewise an equation of the «th degree in jf, in which 
the undetermined coefficients a, j8, y, . . . can so be 
taken as to give the equation in y certain special char- 
acteristics, for example, to make some of the terms 
vanish. From the values of y, the values of x are de- 
termined. By this method the solution of equations 
of the 3rd and 4th degrees is made to depend respec- 
tively upon those of the 2nd and 3rd degrees ; but the 
application of this method to the equation of the 5th 
degree, leads to one of the 24th degree, upon whose 
solution the complete solution of the equation of the 
5th degree depends. 

Afterwards, also, toward the end of the seventeenth 
and the beginning of the eighteenth century, De Lagny, 


Rolle, and Leseur made unsatisfactory attempts to 
advance with rigorous proofs beyond the equation of 
the fourth degree. Euler* took the problem in hand 
in 1749. He attempted first to resolve by means of 
undetermined coefficients the equation of degree 2n 
into two equations each of degree n, but the results 
obtained by him were not more satisfactory than those 
of his predecessors, in that an equation of the eighth 
degree by this treatment led to an equation of the 70th 
degree. These investigations were not valueless, how- 
ever, since through them Eulerf discovered the proof 
of the theorem that every rational integral algebraic 
function of even degree can be resolved into rtal fac- 
tors of the second degree. 

In a work of the date 1762 Euler attacked the so- 
lution of the equation of the nth degree directly. Judg- 
ing from equations of the 2nd and 3rd degrees, he sur- 
mised that a root of the general equation of the nth 
degree might be composed of (n — 1) radicals of the 
nth degree with subordinate square roots. He formed 
expressions of this sort and sought through compari- 
son of coefficients to accomplish his purpose. This 
method presented no difficulty up to the fourth de- 
gree, but in the case of the fifth degree Euler was 
compelled to limit himself to particular cases. For 
example, he obtained from 

-p»_40jc» — 72^« + 50^ + 98=8 
the following value : 

♦ Cantor III., p 582. t Now shown doubtful. 


^=f^— 31 + 3i/=7 ^V—'^l—^\/ZZn 

4.{/_18 + 10t/^+8/— 18 — lOv/I^. 

Analogous to this attempt of Euler is that of War- 
ing (1779). In order to solve the equation /(^) = 
of degree «, he places 

After clearing of radicals, he gets an equation of the 
«th degree, ^(jc) = 0, and by equating coefficients 
finds the necessary equations for determining a, ^, r, 
. . , g and /, but is unable to complete the solution. 

B^zout also proposed a method. He eliminated j' 
from the equations y — 1=0, ay*~^ -{- dy**""^ -{- . . . 
-|-jc = 0, and obtained an equation of the «th degree, 
/(jc)^O, and then equated coeflficients. B^zout was 
no more able to solve the general equation of the 5th 
degree than Waring, but the problem gave him the 
impulse to perfect methods of elimination. 

Tschirnhausen had begun, with his transforma- 
tion, to study the roots of the general equation as func- 
tions of the coeflficients. The same result can be 
reached by another method not different in principle, 
namely the formation of resolvents. In this way, 
Lagrange, Malfatti and Vandermonde independently 
reached results which were published in the year 1771. 
Lagrange's work, rich in matter, gave an analysis of 
all the then known methods of solving equations, and 
explained the difficulties which present themselves in 


passing beyond the fourth degree. Besides this he 
gave methods for determining the limits of the roots 
and the number of imaginary roots, as well as meth- 
ods of approximation. 

Thus all expedients for solving the general equa- 
tion, made prior to the beginning of the nineteenth 
century yielded poor results, and especially with ref- 
erence to Lagrange's work Montiicla* says *'all this 
is well calculated to cool the ardor of those who are 
inclined to tread this new way. Must one entirely 
despair of the solution of this problem?" 

Since the general problem proved insoluble, at- 
tempts were made with special cases, and many ele- 
gant results were obtained in this way. De Moivre 
brought the solution of the equation 

''^+'2^''^+ 2»3'4'5 «y»+....=«, 

for odd integral values of n, into the form 

y=^iya + \/^+l-'i^—a—\/W+l. 

Euler investigated symmetric equations and B^zout 
deduced the relation between the coefficients of an 
equation of the nth degree which must exist in order 
that the same may be transformed into y*'-]-a = 0. 

Gauss made an especially significant step in ad- 
vance in the solution of the cyclotomic equation x^ — 1 
= 0, where « is a prime number. Equations of this 
sort are closely related to the division of the circum- 

* Histoir* eU» Science* Mathitnaiiques, 1799-1802. 


ference into n equal parts. If ^ is the side of an in- 
scribed n-gon in a circle of radius 1, and s the diago- 
nal connecting the first and third vertices, then 

2 IT „ . 2W 

sin—, «=2sin — . 
•^ » n 

If however 

2ir , . . 2,r / 2» , . . 2^^ , 
jf = cos f-^sin — , cos k/sin — =1, 

n « \ « n j 

then the equation x^ — 1 = is to be considered as the 
algebraic expression of the problem of the construc- 
tion of the regular «-gon. 

The following very general theorem was proved 
by Gauss.* '* If « is a prime number, and if « — 1 be 
resolved into prime factors £i, ^, r, . . . so that n — 1 = 
a°-b^ C* , . ., then it is always possible to make the so- 
lution of s^ — 1 = depend upon that of several equa- 
tions of lower degree, namely upon a equations of 
degree «, ^ equations of degree ^, etc." Thus for 
example, the solution of x^ — 1 = (the division of 
the circumference into 73 equal parts) can be effected, 
since n — 1 = 72 = 3^.2', by solving three quadratic 
and two cubic equations. Similarly x^'^ — 1=0 leads 
to four equations of the second degree, since n — 1 = 
16 = 2*; therefore the regular 17-gon can be con- 
structed by elementary geometry, a fact which before 
the time of Gauss no one had anticipated. 

Detailed constructions of the regular 17gon by 
elementary geometry were first given by Pauker and 

* LeKcndre, Thiorie des Nontbres. 


Erchinger.* A noteworthy construction of the same 

figure is due to von Staudt. 

For the case that the prime namber n has the form 2"'-|-l, 
the solution of the equation x" — 1 as depends upon the solution 
of m quadratic equations of which only m — 1 are necessary in the 
construction of the regular »-gon. It should be observed that for 
i» = 2* (A a positive integer), the number 2*" + l may be prime, 
but, as Euler (1738) first pointed out, is not necessarily prime.f If 
m is given successively the values 

1, 2, 8. 4, 6. 6. 7, 8, 16. 2^*, 2«», 

n =B 2"" -|- 1 will take the respective values 

8, 5, 9, 17, 88, 65, 129, 257. 65587. 2«" +1, 2«*' + 1. 

of which only 8. 5. 17. 257, 65537 are prime. The remaining num- 
bers are composite ; in particular, the last two values of n have 
respectively the factors 114689 and 167 772161. The circle there- 
fore can be divided into 257 or 65537 equal parts by solving re- 
spectively 7 or 15 quadratic equations, which is possible by ele- 
mentary geometric construction. 
From the equalities 

255 = 2« — 1=(2* — 1)(2* + 1)= 1517, 256=2*. 
65535 = 2" — 1 = (2« — 1) {2^ -f 1) = 255 • 257, 65536 =» 2>«, 

it is easily seen that, by elementary geometry, that is, by use of 
only straight edge and compasses, the circle can be divided respec- 
tively into 255. 256, 257 ; 65535, 65536, 65537 equal parts. The 
process cannot be continued without a break, since » = 2'' -|- 1 is 
not prime. 

The possibility of an elementary geometric construction of the 
regular 65585-gon is evident from the following : 

65535 = 255 • 257 = 15 • 17 • 257. 

If the circumference of the circle is 1, then since 

* Gauss, lVerke\ II., p. 187. 

t Netto. SubttHuiionentheorir. 1882 ; English by Cole, 1892, p. 187. 

ALGEBRA. . 163 

it follows that av^w of the circumference can be obtained by ele- 
mentary geometric operations. 

After Gauss had given in his earliest scientific 
work, his doctor's dissertation, the first of his proofs 
of the important theorem that every algebraic equa- 
tion has a real or an imaginary root, he made in his 
great memoir of 1801 on the theory of numbers, the 
conjecture that it might be impossible to solve gen- 
eral equations of degree higher than the fourth by 
radicals. Ruffini and Abel gave a rigid proof of this 
fact, and it is due to these investigations that the 
fruitless efforts to reach the solution of the general 
equation by the algebraic method were brought to an 
end. In their stead the question formulated by Abel 
came to the front, ** What are the equations of given 
degree which admit of algebraic solution?" 

The cyclotomic equations of Gauss form such a 
group. But Abel made an important generalization 
by the theorem that an irreducible equation is always 
soluble by radicals when of two roots one can be ra- 
tionally expressed in terms of the other, provided at 
the same time the degree of the equation is prime ; if 
this is not the case, the solution depends upon the 
solution of equations of lower degree. 

A further great group of algebraically soluble equa- 
tions is therefore comprised in the Abelian equations. 
But the question as to the necessary and sufficient 
conditions for the algebraic solubility of an equation 


was first answered by the youthful Galois, the crown 
of whose investigations is the theorem, **I£ the degree 
of an irreducible equation is a prime number, the 
equation is soluble by radicals alone, provided the 
roots of this equation can be expressed rationally in 
terms of any two of them." 

Abel's investigations fall between the years 1824 
and 1829, those of Galois in the years 1830 and 1831. 
Their fundamental significance for all further labors 
in this direction is an undisputed fact ; the question 
concerning the general type of algebraically soluble 
equations alone awaits an answer. 

Galois, who also earned special honors in the field 
of modular equations which enter into the theory of 
elliptic functions, introduced the idea of a group of 
substitutions.* The importance of this innovation, 
and its development into a formal theory of substitu- 
tions, as Cauchy has first given it in the Exercices 
d^ analyse i etc., when he speaks of ** systems of con- 
jugate substitutions,'' became manifest through geo- 
metric considerations. The first example of this was 
furnished by Hesse f in his investigation on the nine 
points of inflexion of a curve of the third degree. The 
equation of the ninth degree upon which they depend 
belongs to the class of algebraically soluble equations. 
In this equation there exists between any two of the 
roots and a third determined by them an algebraic re- 

*Netto, Smbsittuttonentkeorie, 1882. English by Cole, 1892. 
tNoether, Schldniilch*s Zeitschri/t, Band 20. 


lation expressing the geometric fact that the nine 
points of inflexion lie by threes on twelve straight 
lines. To the development of the substitution theory 


in later times, Kronecker, Klein, Noether, Hermite, 
Betti, Serret, Poincar^, Jordan, Capelli, and Sylow 
especially have contributed. 

Most of the algebraists of recent times have par- 
ticipated in the attempt to solve the equation of the 
fifth degree. Before the impossibility of the algebraic 
solution was known, Abel at merely sixteen had made 
an attempt in this direction ; but an essential advance 
is to be noted from the time when the solution of the 
equation of the fifth degree was linked with the theory 
of elliptic functions.* By the help of transformations 
as given on the one hand by Tschirnhausen and on 
the other by E. S. Bring (1786), the roots of the equa- 
tion of the fifth degree can be made to depend upon 
a single quantity only, and therefore the equation, as 
shown by Hermite, can be put into the form fi — / — A 
= 0. By Riemann's methods, the dependence of the 
roots of the equation upon the parameter A is illus- 
trated; on the other hand, it is possible by power- 
series to calculate these five roots to any degree of ap- 
proximation. In 1858, Hermite and Kronecker solved 
the equation of the fifth degree by elliptic functions, 
but without reference to the algebraic theory of this 
equation, while Klein gave the simplest possible solu- 

♦ Klein, F., VergUichende BetrachtHngen Uber neuere geametrischt For- 
tchungen, 1872. 


tion by transcendental functions by using the theory 
of the icosahedron. 

The solution of general equations of the nth degree for ff>4 
by transcendental functions has therefore become possible, and 
the operations entering into the solution are the following : Solu- 
tion of equations of lower degree ; solution of linear differential 
equations with known singular points ; determination of constants 
of integration, by calculating the moduli of periodicity of hyper- 
elliptic integrals for which the branch-points of the function to be 
integrated are known ; finally the calculation of theta-functions of 
several variables for special values of the argument. 

The methods leading to the complete solution of 
an algebraic equation are in many cases tedious ; on 
this account the methods of approximation of real 
roots are very important, especially where they can 
be applied to transcendental equations. The most 
general method of approximation is due to Newton 
(communicated to Barrow in 1669), but was also 
reached by Halley and Raphson in another way.* 
For the solution of equations of the third and fourth 
degrees, John Bernoulli worked out a valuable method 
of approximation in his Lectiones calculi integralis. 
Further methods of approximation are due to Daniel 
Bernoulli, Taylor, Thomas Simpson, Lagrange, Le- 
gendre, Horner, and others. 

By graphic and mechanical means also, the roots of an equa- 
tion can be approximated. C. V Boysf made use of a machine 
for this purpose, which consisted of a system of levers and ful- 
crums ; Cunynghamef used a cubic parabola with a tangent scale 

•Montucla. ^Nature, XXXIII.. p. i66. 


on a straight edge ; C. Reuschle* used an hyperbola with an ac- 
companying gelatine-sheet, so that the roots could be read as in- 
tersections of an hyperbola with a parabola. Similar methods, 
suited especially to equations of the third and fourth degrees are 
due to Bartl, R. Hoppe, and Oekinghausf ; Lalanne and Mehmke 
also deserve mention in this connection. 

For the solution of equations, there had been in- 
vented in the seventeenth century an algorism which 
since then has gained a place in all branches of mathe- 
matics, the algorism of determinants. J The first sug- 
gestion of computation with those regularly formed 
aggregates, which are now called determinants (after 
Cauchy), was given by Leibnitz in the year 1693. 
He used the aggregate 

«ii> ^i«> ^J^lw 

^If ^Mj tfi« 

in forming the resultant of n linear equations with 
n — 1 unknowns, and that of two algebraic equations 
with one unknown. Cramer (1750) is considered as 
a second inventor, because he began to develop a sys- 
tem of computation with determinants. Further the- 
orems are due to B^zout (1764), Vandermonde (1771), 
Laplace (1772), and Lagrange (1773). Gauss's £>is- 
quisUiones arithmeticae (1801) formed an essential ad- 

* BOklen, O., Maik. Mittheilungen^ 1886, p. lot. 

*( Fortschrittt, 1883; 1884. 

X Muir, T., Thtory 0/ Determinants in the Hittorical Order ef its Dez'elop- 
ment. Parti, 1890; Baltzer. R., Theorie und Anwendungen der Determinanten, 


vance, and this gave Cauchy the impulse to many 
new investigations, especially the development of the 
general law (1812) of the multiplication of two deter- 

Jacobi by his ** masterful skill in technique,** also 
rendered conspicuous service in the theory of determi- 
nants, having developed a theory of expressions which 
he designated as ''functional determinants." The 
analogy of these determinants with differential quo- 
tients led him to the general ** principle of the last 
multiplier" which plays a part in nearly all problems 
of integration.* Hesse considered in an especially 
thorough manner symmetric determinants whose ele- 
ments are linear functions of the co-ordinates of a 
geometric figure. He observed their behavior by lin- 
ear transformation of the variables, and their rela- 
tions to such determinants as are formed from them 
by a single bordering. *}* Later discussions are due to 
Cayley on skew determinants, and to Nachreiner and 
S. Giinther on relations between determinants and 
continued fractions. 

The appearance of the differential calculus forms 
one of the most magnificent discoveries of this period. 
The preparatory ideas for this discovery appear in 
manifest outline in Cavalieri,! who in a work Metho 
dus indivisibilium (1635) considers a space-element as 

* Dirichlet, " Ged&chtnissrede anf Jacobi." CrelU't Journal^ Band 52. 
t Noether, Schlbmilch's Zeitschrift, Band 20. 

tLiiroth, Rektoratsrede, Freiburg, 1889; Cantor, II., p. 759. 


the sum of an infinite number of simplest space-ele- 
ments of the next lower dimension, e. g. , a solid as 
the sum of an infinite number of planes. The danger 
of this conception was fully appreciated by the inven- 
tor of the method, but it was improved first by Pascal 
who considers a surface as composed of an infinite 
number of infinitely small rectangles, then by Fermat 
and Roberval ; in all these methods, however, there 
appeared the drawback that the sum of the resulting 
series could seldom be determined. Kepler remarked 
that a function can vary only slightly in the vicinity 
of a greatest or least value. Fermat, led by this 
thought, made an attempt to determine the maximum 
or minimum of a function. Roberval investigated the 
problem of drawing a tangent to a curve, and solved 
it by generating the curved line by the composition of 
two motions, and applied the parallelogram of veloci- 
ties to the construction of the tangents. Barrow, 
Newton's teacher, used this preparatory work with 
reference to Cartesian co-ordinate geometry. He 
chose the rectangle as the velocity- parallelogram, and 
at the same time introduced like Fermat infinitely 
small quantities as increments of the dependent and 
independent variables, with special symbols. He gave 
also the rule, that, without affecting the validity of the 
result of computation, higher powers of infinitely small 
quantities may be neglected in comparison with the 
first power. But Barrow was not able to handle frac- 
tions and radicals involving infinitely small quantities, 


and was compelled to resort to transformations to re- 
move them. Like his predecessors, he was able to 
determine in the simpler cases the value of the quo- 
tient of two, or the sum of an infinite number of in- 
finitesimals. The general solution of such questions 
was reached by Leibnitz and Newton, the founders of 
the differential calculus. 

Leibnitz gave for the calculus of infinitesimals, the 
notion of which had been already introduced, further 
examples and also rules for more complicated cases. 
By summation according to the old methods,* he de- 
duced the simplest theorems of the integral calculus, 
which he, by prefixing a long S as the sign of. summa- 
tion wrote, 

From the fact that the sign of summation f raised 
the dimension, he drew the conclusion that by differ- 
ence-forming the dimension must be diminished so 
that, therefore, as he wrote in a manuscript of Oct. 

29, 1675, from f/^szya, follows immediately ^='^« 

Leibnitz tested the power of his new method by 
geometric problems; he sought, for example, to de- 
termine the curve ** for which the intercepts on the 
axis to the feet of the normals vary as the ordinates." 
In this he let the abscissas x increase in arithmetic 
ratio and designated the constant difference of the 

• Gerhardt, Gtackickte der Mathemaiik in DeuischUnd, 1877; Cantor, III., 
p« x6o. 


abscissas first by -=- and later by dxy without explain- 
ing in detail the meaning of this new symbol. In 
1676 Leibnitz had developed his new calculus so far 
as to be able to solve geometric problems which could 
not be reduced by other methods. Not before 1686, 
however, did he publish anything about his method, 
its great importance being then immediately recog- 
nized by Jacob Bernoulli. 

What Leibnitz failed to explain in the develop- 
ment of his methods, namely what is understood by 
his infinitely small quantities, was clearly expressed 
by Newton, and secured for him a theoretical superi- 
ority. Of a quotient of two infinitely small quantities 
Newton speaks as of a limiting value * which the ratio 
of the vanishing quantities approaches, the smaller 
they become. Similar considerations hold for the sum 
of an infinite number of such quantities. For the de- 
termination of limiting values, Newton devised an 
especial algorism, the calculus of fluxions, which is 
essentially identical with Leibnitz's differential calcu- 
lus. Newton considered the change in the variable 
as a flowing ; he sought to determine the velocity of 
the variation of the function when the variable changes 
with a given velocity. The velocities were called 
fluxions and were designated by jc, y^ % (instead of 
dx, dy, dz, as in Leibnitz's writings). The quantities 
themselves were called fluents, and the calculus of 
fluxions determines therefore the velocities of given 

* Lfiroth, Rtkt^atsrede, Freibar^, 1889. 


motions, or seeks conversely to find the motions when 
the law of their velocities is known. Newton's paper 
on this subject was finished in 1671 under the name 
of Meihodus fiuxionufity but was first published in 1736, 
after his death. Newton is thought by some to have 
borrowed the idea of fluxions from a work of Napier.* 
According to Gauss, Newton deserved much more 
credit than Leibnitz, although he attributes to the 
latter great talent, which, however, was too much dis- 
sipated. It appears that this judgment, looked at 
from both sides, is hardly warranted. Leibnitz failed 
to give satisfactory explanation of that which led 
Newton to one of his most important innovations, the 
idea of limits. On the other hand, Newton is not 
always entirely clear in the purely analytic proof. 
Leibnitz, too, deserves very high praise for the intro- 
duction of the appropriate symbols C and dx^ as well 
as for stating the rules of operating with them. To- 
day the opinion might safely be expressed that the 
differential and integral calculus was independently 
discovered by Newton and by Leibnitz ; that Newton 
is without doubt the first inventor ; that Leibnitz, on 
the other hand, stimulated by the results communi- 
cated to him by Newton, but without the knowledge 
of Newton's methods, invented independently the 
calculus; and that finally to Leibnitz belongs the 
priority of publication. " f 

* Cohen, Dtu Prtnuip der Infinitenmalnuthcde uud seine Getckiekie, 1889; 
Cantor, III., p. 163. 

tLiiroth. A very good summary of the discussion is also given in Ball's 

ALGEBRA. 1 73 

The systematic development of the new calculus 
made necessary a clearer understanding of the idea of 
the infinite. Investigations on the infinitely great are 
of course of only passing interest for the explanation 
of natural phenomena,* but it is entirely different 
with the question of the infinitely small. The infini- 
tesimal f appears in the writings of Kepler as well as 
in those of Cavalieri and Wallis under varying forms, 
essentially as ''infinitely small null value," that is, as 
a quantity which is smaller than any given quantity, 
and which forms the limit of a given finite quantity. 
Euler's indivisibilia lead systematically in the same 
direction. Fermat, Roberval, Pascal, and especially 
Leibnitz and Newton operated with the **unlimitedly 
small," yet in such a way that frequently an abbrevi- 
ated method of expression concealed or at least ob- 
scured the true sense of the development. In the 
writings of John Bernoulli, De THospital, and Pois- 
son, the infinitesimal appears as a quantity difierent 
from zero, but which must become less than an assign- 
able value, i. e.', as a '' pseudo-infinitesimal *' quantity. 
By the formation of derivatives, which in the main 
are identical with Newton's fluxions, Lagrange J at- 
tempted entirely to avoid the infinitesimal, but his 
attempts only served the purpose of bringing into 

Short History of Maihtmaticx^ London, 1888. The best summary it that given 
in Cantor, Vol. III. 

* Riemann, JVerke^ p. 267. 

t R. Hoppe, Differentialrechnung^ 1865. 

(Liiroth, Rektoratsrede^ Freiburg, 1889. 


prominence the urgent need for a deeper foundation 
for the theory of the infinitesimal for which Tacquet 
and Pascal in the seventeenth century, and Maclaurin 
and Carnot in the eighteenth had made preparation. 
We are indebted to Cauchy for this contribution. In 
his investigations there is clearly established the mean- 
ing of propositions which contain the expression ** in- 
finitesimal," and a safe foundation for the differential 
calculus is thereby laid. 

The integral calculus was still further extended 
by Cotes, who showed how to integrate rational alge- 
braic functions. Legendre applied himself to the in- 
tegration of series. Gauss to the approximate deter- 
mination of integrals, and Jacobi to the reduction and 
evaluation of multiple integrals. Dirichlet is espe- 
cially to be credited with generalizations on definite 
integrals, his lectures showing his great fondness for 
this theory.* He it was who welded the scattered 
results of his predecessors into a connected whole, 
and enriched them by a new and original method of 
integration. The introduction of a discontinuous fac- 
tor allowed him to replace the given limits of integra- 
tion by different ones, often by infinite limits, without 
changing the value of the integral. In the more re- 
cent investigations the integral has become the means 
of defining functions or of generating others. 

In the realm of differential equations f the works 

* Kummer, " Ged&chtnissrede anf Lejeune-Dirichlet." Berliner Abh., i860. 
t Cantor, III., p. 429; Schlesinger, L.. Handhuch dtr Theorie der Unearen 


worthy of note date back to Jacob and John Bernoulli 
and to Riccati. Riccati applied Newton's methods 
to the study of th^ problems of the material universe. 
He also integrated for special cases the differential 
equation named in his honor — an equation completely 
solved by Daniel Bernoulli — and discussed the ques- 
tion of the possibility of lowering the order of a given 
differential equation. The theory first received a de- 
tailed scientific treatment at the hands of Lagrange, 
especially as far as concerns partial differential equa- 
tions, of which D'Alembert and Euler had handled 

d u d u 
the equation -^ = -i-^' Laplace also wrote on this 

differential equation and on the reduction of the solu- 
tion of linear differential equations to definite integ- 

On German soil, J. F. Pfaff, the friend of Gauss 
and next to him the most eminent mathematician 
of that time, presented certain elegant investigations 
(1814, 1815) on differential equations,* which led 
Jacobi to introduce the name *'Pfafl&an problem." 
Pfaff found in an original way the general integration 
of partial differential equations of the first degree for 
any number of variable quantities. Beginning with 
the theory of ordinary differential equations of the 
first degree with n variables, for which integrations 

Differ entialgleichungen^ Bd. I., 1895,— an excellent historical review; Mansion. 
P., Theorie der pariiellen Differeniialgleichungen erster Ordnungy deutsch 
von Maser, Leipzig, 1892, also excellent on history. 

*A. Brill, "Das mathematisch-physikalische Seminar in Tiibingen." 
Aus der Festschrift der Universitat Mum Kifm'gs-Judildum, 1889. 


were given by Monge (1809) in special simple cases, 
Pfafi gave their general integration and considered 
the integration of partial differencial equations as a 
particular case of the general integration. In this the 
general integration of differential equationsTof every 
degree between two variables is assumed as known.* 
Jacobi (1827, 1836) also advanced the theory of differ- 
ential equations of the first order. The treatment 
was so to determine unknown functions that an integ- 
ral which contains these functions and the differentia] 
coefficient in a prescribed way reaches a maximum or 
minimum. The condition therefor is the vanishing of 
the first variation of the integral, which again finds its 
expression in differential equations, from which the 
unknown functions are determined. In order to be 
able to distinguish whether a real maximum or mini- 
mum appears, it is necessary to bring the second va- 
riation into a form suitable for investigating its sign. 
This leads to new differential equations which La- 
grange was not able to solve, but of which Jacobi was 
able to show that their integration can be deduced 
from the integration of differential equations belong- 
ing to the first variation. Jacobi also investigated 
the special case of a simple integral with one unknown 
function, his statements being completely proved by 
Hesse. Clebsch undertook the general investigation 
of the second variation, and he was successful in 
showing for the case of multiple integrals that new 

*Gaass, JVkrkf, III., p. 233. 

ALGEBRA. 1 77 

integrals are not necessary for the reduction of the 

second variation. Clebsch (1861, 1862), following the 

suggestions of Jacobi, also reached the solution of the 

Pfaffian problem by making it depend upon a system 

of simultaneous linear partial differential equations, 

the statement of which is possible without integration. 

Of other investigations, one of the most important is 

the theory of the equation 

d^v d^v d^v ^ 

l^'^'d^^^i? * 

which Dirichlet encountered in his work on the po- 
tential, but which had been known since Laplace 
(1789). Recent investigations on differential equa- 
tions, especially on the linear by Fuchs, Klein, and 
Poincar^, stand in close connection with the theories 
of functions and groups, as well as with those of equa- 
tions and series. 

"Within a half century the theory of ordinary differential 
equations has come to be one of the most important branches of 
analysis, the theory of partial differential equations remaining as 
one still to be perfected. The difficulties of the general problem 
of integration are so manifest that all classes of investigators have 
confined themselves to the properties of the integrals in the neigh- 
borhood of certain given points. The new departure took its 
greatest inspiration from two memoirs by Fuchs (1866, 1868), a 
work elaborated by Thom^ and Frobenius. . . . 

"Since 1870 Lie's labors have put the entire theory of differ- 
ential equations on a more satisfactory foundation. He has shown 
that the integration theories of the older mathematicians, which 
had been looked upon as isolated, can by the introduction of the 
concept of continuous groups of transformations be referred to a 


common source, and that ordinary differential equations which 
admit the same infinitesimal transformations present like difficul- 
ties of hitegration He has also emphasized the subject of trans- 
formations of contact (BerUhrunffS'Transformationen) which 
underlies so much of the recent theory. . . . Recent writers have 
shown the same tendency noticeable in the works of Monge and 
Cauchy, the tendency to separate into two schools, the one inclin- 
ing to use the geometric diagram and represented by Schwarz, 
Klein, and Goursat, the other adhering to pure analysis, of which^ 
Weierstrass, Fuchs, and Frobenius are types."* 

A short time after the discovery of the differential 
and integral calculus, namely in the year 1696, John 
Bernoulli proposed this problem to the mathemati- 
cians of his time : To find the curve described by a 
body falling from a given point A to another given 
point B in the shortest time.f The problem came from 
a case in optics, and requires a function to be found 
whose integral is a minimum. Huygens had devel- 
oped the wave-theory of light, and John Bernoulli 
had found under definite assumptions the differential 
equation of the path of the ray of light. Of such mo- 
tion he sought another example, and came upon the 
cycloid as the brachistochrone, that is, upon the above 
statement of the problem, for which up to Easter 
1697, solutions from the Marquis de PHospital, from 
Tschirnhausen, Newton, Jacob Bernoulli and Leib- 
nitz were received. Only the two latter treated the 

♦Smith, D. E., "History of Modern Mathematics," in Merriman and 
Woodward's Higher Mathematics, New York, 1896, with authorities cited. 

t Reiff, R., " Die Anfange der Variationsrechnung," Math, Mittheilungen 
V0H Blfkleu, 1887. Cantor, III., p. 225. Woodhouse, A Treatise on Is^Perimet- 
rical Problems (Cambridge, 1810} . The last named work is rare. 

ALGEBRA. 1 79 

problem as one of maxima and minima. Jacob Ber- 
noulli's method remained the common one for the 
treatment of similar cases up to the time of Lagrange, 
and he is therefore to be regarded as one of the found- 
ers of the calculus of variations. At that time* all 
problems which demanded the statement of a maxi- 
mum or minimum property of functions were called 
isoperimetric problems. To the oldest problems of 
this kind belong especially those in which one curve 
with a maximum or minimum property was to be found 
from a class of curves of equal perimeters. That the 
circle, of all isoperimetric figures, gives the maximum 
area, is said to have been known to Pythagoras. In 
the writings of Pappus a series of propositions on fig- 
ures of equal perimeters are found. Also in the four- 
teenth century the Italian mathematicians had worked 
on problems of this kind. But **the calculus of varia- 
tions may be said to begin with . . . John Bernoulli 
(1696). It immediately occupied the attention of 
Jacob Bernoulli and the Marquis de I'Hospital, but 
Euler first elaborated the subject, "f He| investigated 
the isoperimetric problem first in the analytic- geo- 
metric manner of Jacob Bernoulli, but after he had 
worked on the subject eight years, he came in 1744 
upon a new and general solution by a purely analytic 
method (in his celebrated work : Metliodus inveniendi 

* Ad ton, Getchichte des isoperimetrischen Problems, 1888. 
t Smith, D. E., History of Modern Mathematics, p. 533. 
1: Cantor, III., pp. 243, 819, 830. 


lineas curvas, etc.); this solution shows how those or- 
dinates of the function which are to assume a greatest 
or least value can be derived from the variation of the 
curve-ordinate. Lagrange {Essai d^une nouvelle m^- 
thode^ etc., 1760 and 1761) made the last essential step 
from the pointwise variation of Euler and his prede- 
cessors to the simultaneous variation of all ordinates 
of the required curve by the assumption of variable 
limits of the integral. His methods, which contained 
the new feature of introducing 8 for the change of the 
function, were later taken up in Euler's Integral Cal- 
culus. Since then the calculus of variations has been 
of valuable service in the solution of problems in the- 
ory of curvature. 

The beginnings of a real theory of functions*, espe- 
cially that of the elliptic and Abelian functions lead 
back to Fagnano, Maclaurin, D'Alembert, and Landen. 
Integrals of irrational algebraic functions were treated, 
especially those involving square roots of polynomials 
of the third and fourth degrees; blit none of these 
works hinted at containing the beginnings of a science 
dominating the whole subject of algebra. The matter 
assumed more definite form under the hands of Euler, 
Lagrange, and Legendre. For a long time the only 
transcendental functions known were the circular func- 

* Brill, A., and Noether, M., "Die Entwickelung der Theorie der alge- 
braischen Funcdonen in alterer und neuerer Zeit, Bericht erstattet der Deut- 
schen Mathematiker-Vereinigung, Jahresbericht, Bd. II., pp. 107-566, Berlin, 
1894; Kfinigsberger, L., Zur Gesckichte der Theorie der elliptischen Transcen- 
denten in den Jahren i826-i82g, Leipzig, 1879. 


tions (sin^, cosjtr, . . .)> the common logarithm, and, 
especially for analytic purposes, the hyperbolic log- 
arithm with base ^, and (contained in this) the ex- 
ponential function ^. But with the opening of the 
nineteenth century mathematicians began on the one 
hand thoroughly to study special transcendental func- 
tions, as was done by Legendre, Jacobi, and Abel, 
and on the other hand to develop the general theory 
of functions of a complex variable, in which field 
Gauss, Cauchy, Dirichlet, Riemann, Liouville, Fuchs, 
and Weierstrass obtained valuable results. 

The first signs of an interest in elliptic functions* 
are connected with the determination of the arc of the 
lemniscate, as this was carried out in the middle of 
the eighteenth century. In this Fagnano made the 
discovery that between the limits of two integrals ex- 
pressing the arc of the curve, one of which has twice 
the value of the other, there exists an algebraic rela- 
tion of simple nature. By this means, the arc of the 
lemniscate, though a transcendent of higher order, 
can be doubled or bisected by geometric construc- 
tion like an arc of a circle, "f Euler gave the ex- 
planation of this remarkable phenomenon. He pro- 
duced a more general integral than Fagnano (the 
so-called elliptic integral of the first class) and showed 
that two such integrals can be combined into a third 
of the same kind, so that between the limits of these 

♦Enneper, A., ElliptiscJie Functionen^ Theorie und Geschichte, Halle, 1890. 
t Dirichlet, •' Gedachtnissrede auf Jacobi." CrelU's Journal, Bd. 5a. 


integrals there exists a simple algebraic relation, just 
as the sine of the sum of two arcs can be composed of 
the same functions of the separate arcs (addition- the- 
orem). The elliptic integral, however, depends not 
merely upon the limits but upon another quantity be- 
longing to the function, the modulus. While Euler 
placed only integrals with the same modulus in rela- 
tion, Landen and Lagrange considered those with 
different moduli, and showed that it is possible by 
simple algebraic substitution to change one elliptic 
integral into another of the same class. The estab- 
lishment of the addition-theorem will always remain 
at least as important a service of Euler as his trans- 
formation of the theory of circular functions by the 
introduction of imaginary exponential quantities. 

The origin* of the real theory of elliptic functions 
and the theta-functions falls between 1811 and 1829. 
To Legendre are due two systematic works, the Exer- 
cices de calcul integral (1811-1816) and the Th^orie des 
fonctions elliptiques (1825-1828), neither of which was 
known to Jacobi and Abel. Jacobi published in 1829 
the Fundamenta nova theoriae functionum ellipticaruniy 
certain of the results of which had been simultane- 
ously discovered by Abel. Legendre had recognised 
that a new branch of analysis was involved in those 
investigations, and he devoted decades of earnest 
work to its development. Beginning with the integral 
which depends upon a square root of an expression of 

♦ Cayley. Address to the British Association, etc., 1883. 



the fourth degree in x, Legendre noticed that such 
integrals can be reduced to canonical forms. A^= 
1/1 — ^^sin*^ was substituted for the radical, and 
three essentially different classes of elliptic integrals 
were distinguished and represented by -^(^), ^(^)> 
n(^). These classes depend upon the amplitude ^ 
and the modulus k, the last class also upon a para- 
meter «. 

In spite of the elegant investigations of Legendre 
on elliptic integrals, their theory still presented sev 
eral enigmatic phenomena. It was noticed that the 
degree of the equation conditioning the division of 
the elliptic integral is not equal to the number of the 
parts, as in the division of the circle, but to its square. 
The solution of this and similar problems was re- 
served for Jacobi and Abel. Of the many productive 
ideas of these two eminent mathematicians there are 
especially two which belong to both and have greatly 
advanced the theory. 

In the first place, Abel and Jacobi independently of 
each other observed that it is not expedient to inves- 
tigate the elliptic integral of the first class as a func- 
tion of its limits, but that the method of consideration 
must be reversed, and the limit introduced as a func- 
tion of two quantities dependent upon it. Expressed 
in other words, Abel and Jacobi introduced the direct 
functions instead of the inverse. Abel called them 
^, /, Fy and Jacobi named them sin am^ cos am, A am^ 
or, as they are written by Gudermann, sn, en, dn. 


A second ingenious idea, which belongs to Jacobi 
as well as to Abel, is the introduction of the imagi- 
nary into this theory. As Jacobi himself affirmed, it 
was just this innovation which rendered possible the 
solution of the enigma of the earlier theory. It turned 
out that the new functions partake of the nature of 
the trigonometric and exponential functions. While 
the former are periodic only for real values of the ar- 
gument, and the latter only for imaginary values, the 
elliptic functions have two periods. It can safely be 
said that Gauss as early as the beginning of the nine- 
teenth century had recognised the principle of the 
double period, a fact which was first made plain in 
the writings of Abel. 

Beginning with these two fundamental ideas, Ja- 
cobi and Abel, each in his own way, made further 
important contributions to the theory of elliptic func- 
tions. Legendre had given a transformation of one 
elliptic integral into another of the same form, but a 
second transformation discovered by him was un- 
known to Jacobi, as the latter after serious difficulties 
reached the important result that a multiplication in 
the theory of such functions can be composed of two 
transformations. Abel applied himself to problems 
concerning the division and multiplication of elliptic 
integrals. A thorough study of double periodicity led 
him to the discovery that the general division of the 
elliptic integral with a given limit is always algebraic- 
ally possible as soon a\^^ t! ^e division of the complete 


integrals is assumed as accomplished. The solution 
of the problem was applied by Abel to the lemniscate, 
and in this connection it was proved that the division 
of the whole lemniscate is altogether analogous to 
that of the circle, and can be performed algebraically 
in the same case. Another important discovery of 
AbePs occurred in his allowing, for elliptic functions 
of multiple argument, the multiplier to become infinite 
in formulas deduced from functions with a single ar- 
gument. From this resulted the remarkable expres- 
sions which represent elliptic functions by infinite 
series or quotients of infinite products. 

Jacobi had assumed in his investigations on trans- 
formations that the original variable is rationally ex- 
pressible in terms of the new. Abel, however, entered 
this field with the more general assumption that be- 
tween these two quantities an algebraic equation ex- 
ists, and the result of his labor was that this more 
general problem can be solved by the help of the 
special problem completely treated by Jacobi. 

Jacobi carried still further many of the investiga- 
tions of Abel. Abel had given the theory of the gen- 
eral division, but the actual application demanded 
the formation of certain symmetric functions of the 
roots which could be obtained only in special cases. 
Jacobi gave the solution of the problem so that the 
required functions of the roots could be obtained at 
once and in a manner simpler than AbePs. When 
Jacobi had reached this goal, he stood alone on the 


broad expanse of the new science, for Abel a short 
time before had found an early grave at the age of 27. 

The later efforts of Jacobi culminate in the in- 
troduction of the theta-function. Abel had already 
represented elliptic functions as quotients of infinite 
products. Jacobi could represent these products as 
special cases of a single transcendent, a fact which 
the French mathematicians had come upon in physical 
researches but had neglected to investigate. Jacobi 
examined their analytic nature, brought them into 
connection with the integrals of the second and third 
class, and noticed especially that integrals of the third 
class, though dependent upon three elements, can be 
represented by means of the new transcendent involv- 
ing only two elements. The execution of this process 
gave to the whole theory a high degree of comprehen- 
siveness and clearness, allowing the elliptic functions 
sfiy en, dn to be represented with the new Jacobian 
transcendents <S>i, 02, ©s, 04 as fractions having a com- 
mon denominator. 

What Abel accomplished in the theory of elliptic 
functions is conspicuous, although it was not his 
greatest achievement. There is high authority for 
saying that the achievements of Abel were as great in 
the algebraic field as in that of elliptic functions. But 
his most brilliant results were obtained in the theory 
of the Abelian functions named in his honor, their 
first development falling in the years 1826-1829. 
**Abers Theorem" has been presented by its discov- 


erer in different forms. The paper, Mimoire sur une 
propria^ ginSrale d'une class e tres- it endue de fonctions 
transcendent es^ which after the death of the author re- 
ceived the prize from the French academy, contained 
the most general expression. In form it is a theorem 
of the integral calculus, the integrals depending upon 
an irrational function y^ which is connected with x by 
an algebraic equation F{Xy y)^= 0. Abel's fundamental 
theorem states that a sum of such integrals can be 
expressed by a definite number p of similar integrals 
where / depends only upon the properties of the equa- 
tion F(^x, j')=0. (This / is the deficiency of the curve 
F{x,y)=^^\ the notion of deficiency, however, dates 
first from the year 1857.) For the case that 

y = VAx^-\-B3(^+Cx^-\-Dx-\-E, 

Abel's theorem leads to Legendre's proposition on 
the sum of two elliptic integrals. Here/ = 1. If 

y = l/Ax^ + Bx^+ . . . +P, 

where A can also be 0, then / is 2, and so on. For 
p = 3, or >3, the hyperelliptic integrals are onlysf)e- 
cial cases of the Abelian integrals of like class. 

After Abel's death (1829) Jacobi carried the theory 
further in his Considerationes generates de trans cendenti- 
bus Abelianis (1832), and showed for hyperelliptic in- 
tegrals of a given class that the direct functions to 
which Abel's proposition applies are not functions of 
a single variable, as the elliptic functions j«, cn^ dn^ 
but are functions of p variables. Separate papers of 


essential significance for the case / = 2, are due to 
Rosenhain (1846, published 1851) and Goepel (1847). 
Two articles of Riemann, founded upon the writ- 
ings of Gauss and Cauchy, have become significant 
in the development of the complete theory of func- 
tions. Cauchy had by rigorous methods and by the 
introduction of the imaginary variable **laid the foun- 
dation for an essential improvement and transforma- 
tion of the whole of analysis."* Riemann built upon 
this foundation and wrote the Grundlage fur eine all- 
gemeine Theorie der Funktionen einer verdnderlichen 
komplexen Grosse in the year 1851, and the Theorie der 
AbeVschen Funktionen which appeared six years later. 
For the treatment of the Abelian functions, Riemann 
used theta-functions with several arguments, the the- 
ory of which is based upon the general principle of 
the theory of functions of a complex variable. He 
begins with integrals of algebraic functions of the 
most general form and considers their inverse func- 
tions, that is, the Abelian functions of p variables. 
Then a theta function of / variables is defined as the 
sum of a /-tuply infinite exponential series whose 
general term depends, in addition to / variables, upon 

certain ^^ — - constants which must be reducible 

to 3/ — 3 moduli, but the theory has not yet been com- 

Starting from the works of Gauss and Abel as well 

*Kummer, " Gedftchtnissrede auf Lejeune-Dirichlet,'* Berlintr Abhand- 
iungien, i860. 

ALGEBRA. 1 89 

as the developments of Cauchy on integrations in the 
imaginary plane, a strong movement appears in which 
occur the names of Weierstrass, G. Cantor, Heine, 
Dedekind, P. Du Bois-Reymond, Dini, Scheeffer, 
Pringsheim, Holder, Pincherle, and others. This 
tendency aims at freeing from criticism the founda- 
tions of arithmetic, especially by a new treatment of 
irrationals based upon the theory of functions with its 
considerations of continuity and discontinuity. It 
likewise considers the bases of the theory of series by 
investigations on convergence and divergence, and 
gives to the differential calculus greater preciseness 
through the introduction of mean-value theorems. 

After Riemann valuable contributions to the theory 
of the theta-functions were made by Weierstrass, 
Weber, NOther, H. Stahl, Schottky, and Frobenius. 
Since Riemann a theory of algebraic functions and 
point-groups has been detached from the theory of 
Abelian functions, a theory which was founded through 
the writings of Brill, Nether, and Lindemann upon 
the remainder- theorem and the Riemann -Roch theo- 
rem, while recently Weber and Dedekind have allied 
themselves with the theory of ideal numbers, set forth • 
in the first appendix to Dirichlet. The extremely 
rich development of the general theory of functions 
in recent years has borne fruit in different branches of 
mathematical science, and undoubtedly is to be rec- 
ognised as having furnished a solid foundation for the 
work of the future. 



'T^HE oldest traces of geometry are found among 
^ the Egyptians and Babylonians. In this first 
period geometry was made to serve practical purposes 
almost exclusively. From the Egyptian and Baby- 
lonian priesthood and learned classes geometry was 
transplanted to Grecian soil. Here begins the second 
period, a classic era of {Philosophic conception of geo- 
metric notions as the embodiment of a general science 
of mathematics, connected with the names of Pythag- 
oras, Eratosthenes, Euclid, Apollonius, and Archi- 
medes. The works of the last two indeed, touch upon 
lines not clearly defined until modern times. Apollo - 
nius in his Conic Sections gives the first real example 
of a geometry of position, -while Archimedes for the 
most part concerns himself with the geometry of meas- 

The golden age of Greek geometry was brief and 
yet it was not wholly extinct until the memory of the 
great men of Alexandria was lost in the insignificance 
of their successors. Then followed more than a thou- 


sand years of a cheerless epoch which at best was re- 
stricted to borrowing from the Greeks such geometric 
knowledge as could be understood. History might 
pass over these many centuries in silence were it not 
compelled to give attention to these obscure and un- 
productive periods in their relation to the past and 
future. In this third period come first the Romans, 
Hindus, and Chinese, turning the Greek geometry to 
use after their own fashion ; then the Arabs as skilled 
intermediaries between the ancient classic and a mod- 
ern era. 

The fourth period comprises the early develop- 
ment of geometry among the nations of the West. 
By the labors of Arab authors the treasures of a time 
long past were brought within the walls of monasteries 
and into the hands of teachers in newly established 
schools and universities, without as yet forming a 
subject for general instruction. The most prominent 
intellects of this period are Vieta and Kepler. In 
their methods they suggest the fifth period which be- 
gins with Descartes. The powerful methods of analy- 
sis are now introduced into geometry. Analytic geom- 
etry comes into being. The application of its seductive 
methods received the almost exclusive attention of 
the mathematicians of the seventeenth and eighteenth 
centuries. Then in the so-called modern or projective 
geometry and the geometry of curved surfaces there 
arose theories which, like analytic geometry, far tran- 
scended the geometry of the ancients, especially in 


the way of leading to the almost unlimited generaliza- 
tion of truths already known. 



In the same book of Ahmes which has disclosed to 
US the elementary arithmetic of the Egyptians are 
also found sections on geometry, the determination 
of areas of simple surfaces, with figures appended. 
These figures are either rectilinear or circular. Among 
them are found isosceles triangles, * rectangles, isos- 
celes trapezoids and circles.* The area of the rect- 
angle is correctly determined ; as the measure of the 
area of the isosceles triangle with base a and side b, 
however, \ab is found, and for the area of the isosceles 
trapezoid with parallel sides a! and cC' and oblique side 
b, the expression \ {a' -f a") b is given. These approx- 
imate formulae are used throughout and are evidently 
considered perfectly correct. The area of the circle 
follows, with the exceptionally accurate value ir= 

^j =3.1605. 

Among the problems of geometric construction 
one stands forth preeminent by reason of its practical 
importance, viz., to lay off a right angle. The solu- 
tion of this problem, so vital in the construction of 
temples and palaces, belonged to the profession of 

* Cantor, I., p. 52. 


rope- stretchers or harpedonaptae. They used a rope 
divided by knots into three segments (perhaps corre- 
sponding to the numbers 3, 4, 5) iorming a Pythago- 
ean triangle.* 

Among the Babylonians the construction of figures 
of religious significance led up to a formal geometry of 
divination which recognized triangles, quadrilaterals, 
right angles, circles with the inscribed regular hex- 
agon and the division of the circumference into three 
hundred and sixty degrees as well as a value ir=3. 

Stereometric problems, such as finding the con- 
tents of granaries, are found in Ahmes ; but not much 
is to be learned from his statements since no account 
is given of the shape of the storehouses. 

As for projective representations, the Egyptian 
wall sculptures show no evidence of any knowledge 
of perspective. For example a square pond is pic- 
tured in the ground-plan but the trees and the water- 
drawers standing on the bank are added to the picture 
in the elevation, as it were from the outside, f 



In a survey of Greek geometry it will here and 
there appear as if investigations connected in a very 

* Cantor, I., p. 62. 

t Wiener, Lehrbuch tUr darstellenden Geometries 1884. Hereafter referred 
to as Wiener. 


simple manner with well-known theorems were not 
known to the Greeks. At least it seems as if they 
could not have been established satisfactorily, since 
they are thrown in among other matters evidently 
without connection. Doubtless the principal reason 
for this is that a number of the important writings of 
the ancient mathematicians are lost. Another no less 
weighty reason might be that much was handed down 
simply by oral tradition, and the latter, by reason of 
the stiff and repulsive way in which most of the Greek 
demonstrations were worked out, did not always ren- 
der the truths set forth indisputable. 

In Thales are found traces of Egyptian geometry, 
but one must not expect to discover there all that was 
known to the Egyptians. Thales mentions the theo- 
rems regarding vertical angles, the angles at the base 
of an isosceles triangle, the determination of a triangle 
from a side and two adjacent angles, and the angle in- 
scribed in a semi- circle. He knew how to determine 
the height of an object by comparing its shadow with 
the shadow of a staff placed at the extremity of the 
shadow of the object, so that here may be found the 
beginnings of the theory of similarity. In Thales the 
proofs of the theorems are either not given at all or 
are given without the rigor demanded in later times. 

In this direction an important advance was made 
by Pythagoras and his school. To him without ques- 
tion is to be ascribed the theorem known to the Egyp- 
tian "rope stretchers" concerning the right-angled 


triangle, which they knew in the case of the tri 
angle with sides 3, 4, 5, without giving a rigorous 
proof. Euclid's is the earliest of the extant proofs of 
this theorem. Of other matters, what is to be ascribed 
to Pythagoras himself, and what to his pupils, it is 
difficult to decide. The Pythagoreans proved that the 
sum of the angles of a plane triangle is two right an- 
gles. They knew the golden section, and also the 
regular polygons so far as they make up the bound- 
aries of the five regular bodies. Also regular star- 
polygons were known, at least the star-pentagon. In 
the Pythagorean theorems of area the gnomon played 
an important part. This word originally signified the 
vertical staff which by its shadow indicated the hours, 
and later the right angle mechanically represented. 
Among the Pythagoreans the gnomon is the figure 
left after a square has been taken from the corner of 
another square. Later, in Euclid, the gnomon is a 
parallelogram after similar treatment (see page 66). 
The Pythagoreans called the perpendicular to a straight 
line "a line directed according to the gnomon.*'* 

But geometric knowledge extended beyond the 
school of Pythagoras. Anaxagoras is said to have been 
the first to try to determine a square of area equal 
to that of a given circle. It is to be noticed that like 
most of his successors he believed in the possibility 
of solving this problem. CEnopides showed how to 
draw a perpendicular from a point to a line and how 

* Cantor, I., p. 150. 



to lay off a given angle at a given point of a given 
line. Hippias of Elis likewise sought the quadrature 
of the circle, and later he attempted the trisection of 
an angle, for which he constructed the quadratrix. 


This curve is described as follows : Upon a quadrant of a cir- 
cumference cut off by two perpendicular radii, OA and OB, lie 
the points A, . . . JC, L, . . . B. The radius r = OA revolves with 
uniform velocity about O from the position OA to the position OB 
At the same time a straight line ^ always parallel to OA moves 
with uniform velocity from the position OA to that of a tangent to 
the circle at B. If JC' is the intersection of ^ with OB at the time 
when the moving radius falls upon OK" then the parallel to OA 
through fC' meets the radius OK in a point JC" belonging to the 
quadratrix. If F is the intersection of OA with the quadratrix, it 
follows in part directly and in part from simple considerations, that 

arc AK OK' 

arc AL 



a relation which solves any problem of angle sections. 


^^_2r OP _ OA 

n'^^ OA " arcAB' 

whence it is obvious that the quadrature of the circle depends upon 


the ratio in which the radius OA is divided by the point F of the 
quadratrix. If this ratio could be constructed by elementary geom- 
* etry, the quadrature of the circle would be effected.* It appears 
that the quadratrix was first invented for the trisection of an angle 
and that its relation to the quadrature of the circle was discovered 
later, f as is shown by Dinostratus. 

The problem of the quadrature of the circle is also 
found in Hippocrates. He endeavored to accomplish 
his purpose by the consideration of crescent-shaped 
figures bounded by arcs of circles. It is of especial 
importance to note that Hippocrates wrote an ele- 
mentar}^ book of mathematics (the first of the kind) 
in which he represented a point by a single capital 
letter and a segment by t o, although we are unable 
to determine who was the first to introduce this sym- 

Geometry was strengthened on the philosophic 
side by Plato, who felt the need of establishing defini- 
tions and axioms and simplifying the work of the in- 
vestigator by the introduction of the analytic method. 

A systematic representation of the results of all 
the earlier investigations in the domain of elementary 
^geometry, enriched by the fruits of his own abundant 
labor, is given by Euclid in the thirteen books of his 
Elements which deal not only with plane figures but 
also with figures in space and algebraic investiga- 

*The equation of the quadratrix in polar co-ordinates is r« — . — — 
ere a = OA. 

t M ontncla. 

2a ^ si n <^ 

wherea=C?/i. Putting ^=o, r = ro, we have » = 



tions. ."Whatever has been said in praise of mathe- 
matics, of the strength, perspicuity and rigor of its 
presentation, all is especially true of this work of the 
great Alexandrian. Definitions, axioms, and conclu- 
sions are joined together link by link as into a chain, 
firm and inflexible, of binding force but also cold and 
hard, repellent to a productive mind and affording no 
room for independent activity. A ripened understand- 
ing is needed to appreciate the classic beauties of this 
greatest monument of Greek ingenuity. It is not the 
arena for the youth eager for enterprise ; to captivate 
him a field of action is better suited where he Ynay 
hope to discover something new, unexpected."* 

The first book of the Elements deals with the the- 
ory of triangles and quadrilaterals, the second book 
with the application of the Pythagorean theorem to 
a large number of constructions, really of arithmetic 
nature. The third book introduces circles, the fourth 
book inscribed and circumscribed polygons. Propor- 
tions explained by the aid of line segments occupy 
the fifth book, and in the sixth book find their appli- 
cation to the proof of theorems involving the similar 
ity of figures. The seventh, eighth, ninth and tenth 
books have especially to do with the theory of num- 
bers. These books contain respectively the measure- 
ment and division of numbers, the algorism for de- 
termining the least common multiple and the greatest 
common divisor, prime numbers, geometric series, 

*'K. Brill, Aniritisrede in Tubingen, 1884. 


and incommensurable (irrational) numbers. Then 
follows stereometry : in the eleventh book the straight 
line, the plane, the prism ; in the twelfth, the discus- 
sion of the prism, pyramid, cone, cylinder, sphere; 
and in the thirteenth, regular polygons with the regu- 
lar solids formed from them, the number of which 
Euclid gives definitely as five. Without detracting in 
the least from the glory due to Euclid for the compo- 
sition of this imperishable work, it may be assumed 
that individual portions grew out of the well grounded 
preparatory work of others. This is almost> certainly 
true of the fifth book, of which Eudoxus seems to 
have been the real author. 

Not by reason of a great compilation like Euclid, 
but through a series of valuable single treatises, Archi- 
medes is justly entitled to have a more detailed de- 
scription of his geometric productions. In his inves- 
tigations of the sphere and cylinder he assumes that 
the straight line is the shortest distance between two 
points. ' From the Arabic we have a small geometric 
work of Archimedes consisting of fifteen so-called 
lemmas, some of which have value in connection with 
the comparison of figures bounded by straight lines 
and arcs of circles, the trisection of the angle, and 
the determination of cross- ratios. Of especial impor- 
tance is his mensuration of the circle, in which he 
shows w to lie between 3i^ and 3i^^. This as well as 
many other results Archimedes obtains by the method 
of exhaustions which among the ancients usually took 


the place of the modern integration. * The quantity 
sought, the area bounded by a curve, for example, 
may be considered as the limit of the areas of the in- 
scribed and circumscribed polygons the number of 
whose sides is continually increased by the bisection 
of the arcs, and it is shown that the difference between 
two associated polygons, by an indefinite continuance 
of this process, must become less than an arbitrarily 
small given magnitude. This difference was thus, as 
it were, exhausted, and the result obtained by exhaus- 

The field of the constructions of elementary geom- 
etry received at the hands of Apollonius an extension 
in the solution of the problem to construct a circle 
tangent to three given circles, and in the systematic 
introduction of the diorismus (determination or limi- 
tation). This also appears in more difficult problems 
in his Conic. Sections^ from which we see that Apollo- 
nius gives not simply the conditions for the possibility 
of the solution in general, but especially desires to 
determine the limits of the solutions. 

From Zenodorus several theorems regarding iso- 
perimetric figures are still extant ; for example, he 
states that the circle has a greater area than any iso- 
perimetric regular polygon, that among all isoperi- 
metric polygons of the same number of sides the reg- 
ular has the greatest area, and so on. Hypsicles gives 

*Chasles, A^er^u historique sur torigine et le diveloppentent dts fnithodes 
tn giomitrte, 1875. Hereafter referred to as Chasles. 


as something new the division of the circumference 
into three hundred and sixty degrees. From Heron 
we have a book on geometry (according to Tannery 
still another, a commentary on Euclid's Elements) 
which deals in an extended manner with the mensu- 
ration of plane figures. Here we find deduced for the 
area A of the triangle whose sides are a, b, and r, 
where 2s=^a-\-b -\- c, the formula 

A = l/j'(x — a){s — b){s—c). 

In the measurement of the circle we usually find ^ as 
an approximation for ir; but still in the Book of Meas- 
urements we also find 7r = 3. 

In the period after the commencement of the 
Christian era the output becomes still more meager. 
Only occasionally do we find anything noteworthy. 
Serenus, however, gives a theorem on transversals 
which expresses the fact that a harmonic pencil is cut 
by an arbitrary transversal in a harmonic range. In 
the Almagest occurs the theorem regarding the in- 
scribed quadrilateral, ordinarily known as Ptolemy's 
Theorem, and a value written in sexagesimal form 
7r = 3.8.30, i. e. , 

'' = ^+ 60 + 60-60 =^T20 = ^-^^^^^ • • • • * 

In a special treatise on geometry Ptolemy shows that 
he does not regard Euclid's theory of parallels as in- 

* Cantor, I., p. 394. 


To the last supporters of Greek geometry belong 
Sextus Julius Africanus, who determined the width of 
a stream by the use of similar right-angled triangles, 
and Pappus, whose name has become very well known 
by reason of his Collection. This work consisting orig- 
inally of eight books, of which the first is wholly 
lost and the second in great part, presents the sub- 
stance of the mathematical writings of special repute 
in the time of the author, and in some places adds 
corollaries. Since his work was evidently composed 
with great conscientiousness, it has become one of 
the most trustworthy sources for the study of the 
mathematical history of ancient times. The geomet- 
ric part of the Collection contains among other things 
discussions of the three different means between two 
line-segments, isoperimetric figures, and tangency of 
circles. It also discusses similarity in the case of cir- 
cles ; so far at least as to show that all lines which 
join the ends of parallel radii of two circles, drawn in 
the same or in opposite directions, intersect in a fixed 
point of the line of centers. 

The Greeks rendered important service not simply 
in the field of elementary geometry : they are also the 
creators of the theory of conic sections. And as in 
the one the name of Euclid, so in the other the name 
of Apollonius of Perga has been the signal for con- 
troversy. The theory of the curves of second order 
does not begin with Apollonius any more than does 
Euclidean geometry begin with Euclid ; but what the 


Elements signify for elementary geometry, the eight 
books of the Conies signify for the theory of lines of 
the second order. Only the first four books of the 
Conic Sections of Apollonius are preserved in the 
Greek text : the next three are known through Arabic 
translations : the eighth book has never been found 
and is given up for lost, though its contents have been 
restored by H alley from references in Pappus. The 
first book deals with the formation of conies by plane 
sections of circular cones, with conjugate diameters, 
and with axes and tangents. The second has espe- 
cially to do with asymptotes. These Apollonius ob- 
tains by laying off on a tangent from the point of con- 
tact the half-length of the parallel diameter and joining 
its extremity to the center of the curve. The third 
book contains theorems on foci and secants, and the 
fourth upon the intersection of circles with conies and 
of conies with one another. With this the elementary 
treatment of conies by Apollonius closes. The fol- 
lowing books contain special investigations in applica- 
tion of the methods developed in the first four books. 
Thus the fifth book deals with the maximum and min- 
imum lines which can be drawn from a point to the 
conic, and also with the normals from a given point 
in the plane of the curve of the second order; the sixth 
with equal and similar conies ; the seventh in a re- 
markable manner with the parallelograms having con- 
jugate diameters as sides and the theorem upon the 
sum of the squares of conjugate diameters. The eighth 


book contained, according to Halley, a series of prob- 
lems connected in the closest manner with lemmas of 
the seventh book. 

The first effort toward the development of the the- 
ory of conic sections is ascribed to Hippocrates.* He 
reduced the duplication of the cube to the construc- 
tion of two mean proportionals x andy between two 
given line-segments a and d ; thusf 

— = — = ^ gives x^ == ay, y^ = bx, whence 

x^z=a^b=^ — a^ =m'(jfi, 

Archytas and Eudoxus seem to have found, by plane 
construction, curves satisfying the above equations 
but different from straight lines and circles. Menaech- 
mus sought for the new curves, already known by 
plane constructions, a representation by sections of 
cones of revolution, and became the discoverer of 
conic sections in this sense. He employed only sec- 
tions perpendicular to an element of a right circular 
cone; thus the parabola was designated as the ** sec- 
tion of a right-angled cone*' (whose generating angle 
is 45°) ; the ellipse, the " section of an acute-angled 
cone*'; the hyperbola, the ** section of an obtuse- 
angled cone." These names are also used by Archi- 
medes, although he was aware that the three curves 
can be formed as sections of any circular cone. Apol 

*Zeutben, Die Lehre von den Kegelschnitten itn Aliertum. Deutsch vou 
V. Fischer-Benzon, 1886. P. 459. Hereafter referred to as Zeuthen. 

t Cantor, I., p> 200. 



lonius first introduced the names ** ellipse,' "para- 
bola," "hyperbola.** Possibly Menaechmus, but in 
any case Archimedes, determined conies by a linear 
equation between areas, of the form y^=zkxx\. The 
semi-parameter, with Archimedes and possibly some 
of his predecessors, was known as "the segment to 
the axis," i. e., the segment of the axis of the circle 
from the vertex of the curve to its intersection with 
the axis of the cone. The designation "parameter" 
is due to Desargues (1639).* 

It has been shown f that ApoUonius represented the conies by 
equations of the form ^=^fx-\-ax^, where jc and jy are regarded 
as parallel coordinates and every term is represented as an area. 
From this other linear equations involving areas were derived, and 
so equations belonging to analytic geometry were obtained by the 
use of a system of parallel coordinates whose origin could, for 
geometric reasons, be shifted simultaneously with an interchange 
of axes. Hence we already find certain fundamental ideas of the 
analytic geometry which appeared almost two thousand years later. 

The study of conic sections was continued upon the 
cone itself only till the time when a single fundamen- 
tal plane property rendered it possible to undertake 
the further investigation in the plane. J In this way 
there had become known, up to the time of Archi- 
medes, a number of important theorems on conjugate 
diameters, and the relations of the lines to these di- 
ameters as axes, by the aid of linear equations be- 

*BaItzer, R., Analyttscke Geometrie, 1882. 
t Zeuthen, p. 32. X Zeuthen, p. 43. 


tween areas. Tliere were also known the so-called 
Newton's power theorem, the theorem that the rect- 
angles of the segments of two secants of a conic drawn 
through an arbitrary point in given direction are in a 
constant ratio, theorems upon the generation of a 
conic by aid of its tangents or as the locus related to 
four straight lines, and the theorem regarding pole 
and polar. But these theorems were always applied 
to only one branch of the hyperbola. One of the valu- 
able services of Apollonius was to extend his own 
theorems, and consequently those already known, to 
both branches of the hyperbola. His whole method 
justifies us in regarding him the most prominent rep- 
resentative of the Greek theory of conic sections, and 
so much the more when wej can see from his principal 
work that the foundations for the theory of projective 
ranges and pencils had virtually been laid by the an- 
cients in different theorems and applications. 

With Apollonius the period of new discoveries in 
the realm of the theory of conies comes to an end. In 
later times we find only applications of long known 
theorems to problems of no great difficulty. Indeed, 
the Solution of problems already played an important 
part in the oldest times of Greek geometry and fur- 
nished the occasion for the exposition not only of 
conies but also of curves of higher order than the sec- 
ond. In the number of problems, which on account 
of their classic value have been transmitted from gen- 
eration to generation and have continually furnished 


occasion for further investigation, three, by reason of 
their importance, stand forth preeminent : the duplica- 
tion of the cube, or more generally the multiplication 
of the cube, the trisection of the angle and the quad- 
rature of the circle. The appearance of these three 
problems has been of the greatest significance in the 
development of the whole of mathematics. The first 
requires the solution of an equation of the third de- 
gree ; the second (for certain angles at least) leads to 
an important section of the theory of numbers, i. e., 
to the cyclotomic equations, and Gauss (see p. i6o) 
was the first to show that by a finite number of ope- 
rations with straight edge and compasses we can con- 
struct a regular polygon of n sides whenever « is a 
prime number and n — 1=22^. The third problem 
reaches over into the province of algebra, for Linde- 
mann* in the year 1882 showed that ir cannot be the 
root of an algebmic equation with integral coefficients. 
The multiplication of the cube, algebraically the 
determination of x from the equation 


is also called the Delian problem, because the Delians 
were required to double their cubical altar, f The so- 
lution of this problem was specially studied by Plato, 
Archytas, and Menaechmus; the latter solved it by 

*Mathent, Annalen, XX., p. 215. See also Mathent. Annalen^ XLIII., and 
Klein, Famous Problems of Elementary Geometry^ 1895, translated by Beman 
and Smith, Boston, 1897. 

t Cantor, I., p. 2x9. 

:k:' = — cfi=ma^t 


the use of^ conies (hyperbolas and parabolas). Era- 
tosthenes constructed a mechanical apparatus for the 
same purpose. 

Among the solutions of the problem of the trisec- 
tion of an angle, the method of Archimedes is note- 
worthy. It furnishes an example of the so called 
** insertions" of which the Greeks made use when a 
solution by straight edge and compasses was impos- 
sible. His process was as follows : Required to divide 
the arc AB of the circle with center M into three 
equal parts. Draw the diameter AE, and through B 
a secant cutting the circumference in C and the di- 
ameter AE in Z>, so that CD equals the radius r of 
the circle. Then arc CE = \AB. 


According to the rules of insertion the process con- 
sists in laying off upon a ruler a length r, causing it 
to pass through B while one extremity D of the seg- 
ment r slides along the diameter AE. By moving 
the ruler we get a certain position in which the other 
extremity of the segment r falls upon the circumfer- 
ence, and thus the point C is determined. 

This problem Pappus claims to have solved after 
the manner of the ancients by the use of conic sec- 


tions. Since in the writings of Apollonius, so largely 
lost, lines of the second order find an extended appli- 
cation to the solution of problems, the conies were 
frequently called solid loci in opposition to plane loci, 
i. e., the straight line and circle. Following these 
came lihear loci, a term including all other curves, of 
which a large number were investigated. 

This designation of the loci is found, for example, 
in Pappus, who says in his seventh book * that a prob- 
lem is called plane, solid, or linear, according as its 
solution requires plane, solid, or linear loci. It is, 
however, highly probable that the loci received their 
names from problems, and that therefore the division 
of problems into plane, solid, and linear preceded the 
designation of the corresponding loci. First it is to 
be noticed that we do not hear of ** linear problems 
and loci" till after the terms ** plane and solid prob- 
lems and loci" were in use. Plane problems were 
those which in the geometric treatment proved to be 
dependent upon equations of the first or second de 
gree between segments, and hence could be solved 
by the simple application of areas, the Greek method 
for the solution of quadratic equations. Problems de- 
pending upon the solution of equations of the third 
degree between segments led to the use of forms of 
three dimensions, as, e. g., the duplication of the 
cube, and were termed solid problems; the loci used 
in their solution (the conies) were solid loci. At a 

♦Zeuthen, p. 203. 


time when the significance of ''plane" and **solid" 
was forgotten, the term "linear problem" was first 
applied to those problems whose treatment (by ''lin- 
ear loci") no longer led to equations of the first, sec- 
ond, and third degrees, and which therefore could no 
longer be represented as linear relations between seg- 
ments, areas, or volumes. 

Of linear loci Hippias applied the quadratrix (to 
which the name of Dinostratus was later attached 
through his attempt at the quadrature of the circle)* 
to the trisection of the angle. Eudoxus was acquainted 
with the sections of the torus made by planes parallel 
to the axis of the surface, especially the hippopede or 
figure of- eight curve, f The spirals of Archimedes 
attained special celebrity. His exposition of their 
properties compares favorably with his elegant inves- 
tigations of the quadrature of the parabola. 

Conon had already generated the spiral of Archi- 
medes J by the motion of a point which recedes with 
uniform velocity along the radius OA of a circle k 
from the center (?, while OA likewise revolves uni- 
formly about O, But Archimedes was the first to dis- 
cover certain of the beautiful properties of this curve; 
he found that if, after one revolution, the spiral meets 
the circle k of radius OA in B (where BO is tangent 
to the spiral at O), the area bounded by BO and the 

* Cantor, I., pp. 184, 233. 

t Majer, Proklos Uber die Petita und Axiomata bet Euklid, 1875. 

t Cantor, I., p. 291. 


Spiral is one-third of the area of the circle k; further 
that the tangent to the spiral at B cuts off from a per- 
pendicular to OB at 6> a segment equal to the circum- 
ference of the circle k.* 

The only noteworthy discovery of Nicomedes is 
the construction of the conchoid which he employed 
to solve the problem of the two mean proportionals, 
or, what apiounts to the same thing, the multiplica- 
tion of the cube. The curve is the geometric locus of 
the point X upon a moving straight line g which con- 
stantly passes through a fixed point P and cuts a fixed 
straight line ^ in Fso that XV has a constant length. 
Nicomedes also investigated the properties of this 
curve and constructed an apparatus made of rulers 
for its mechanical description. 

The cissoid of Diodes is also of use in the multi- 
plication of the cube. It may be constructed as fol- 
lows : Through the extremity A of the radius OA of 
a circle k passes the secant ^C which cuts k in C and 
the radius OB perpendicular to OA in D; X, upon 
^C, is a point of the cissoid when £>X=£>C.1[ Gemi- 
nus proves that besides the straight line and the circle 
the helical curve invented by Archytas possesses the 
insertion property. 

Along with the geometry of the plane was devel- 
oped the geometry of space, first as elementary stere- 

* Montncla. 

+ Klein, F., Famous ProbL'tns of Elementary Geometry, translated by Beman 
nnd Smith, Boston, 1897, P- 44 


ometry and then in theorems dealing with surfaces of 
the second order. The knowledge of the five regular 
bodies and the related circumscribed sphere certainly 
goes back to Pythagoras. According to the statement 
of Timaeus of Locri,* fire is made up of tetrahedra, 
air of octahedra, water of icosahedra, earth of cubes, 
while the dodecahedron forms the boundary of the 
universe. Of these five cosmic or Platonic bodies 
Theaetetus seems to have been the first to publish a 
connected treatment. Eudoxus states that a pyramid 
(or cone) is -^ of a prism of equal base and altitude. 
The eleventh, twelfth and thirteenth books of Euclid's 
Elements offer a summary discussion of the ordinary 
stereometry. (See p. 199.) Archimedes introduces 
thirteen semi-regular solids, i. e., solids whose bound- 
aries are regular polygons of two or three different 
kinds. Besides this he compares the surface and vol- 
ume of the sphere with the corresponding expressions 
for the circumscribed cylinder and deduces theorems 
which he esteems so highly that he expresses the de- 
sire to have the sphere and circumscribed cylinder 
cut upon his tomb-stone. Among later mathemati- 
cians Hypsicles and Herori give exercises in the men- 
suration of regular and irregular solids. Pappus also 
furnishes certain stereometric investigations of which 
we specially mention as new only the determination 
of the volume of a solid of revolution by means of the 
meridian section and the path of its center of gravity. 

* Cantor, I., p. 163. 



He thus shows familiarity with a part of the theorem 
later known as Guldin's rule. 

Of surfaces of the second order the Greeks knew 
the elementary surfaces of revolution, i. e., the sphere, 
the right circular cylinder and circular cone. Euclid 
deals only with cones of revolution, Archimedes on the 
contrary with circular cones in general. In addition, 
Archimedes investigates the ** right-angled conoids" 
(paraboloids of revolution), the <' obtuse-angled co- 
noids" (hyperboloids of revolution of one sheet), and 
"long and flat spheroids" (ellipsoids of revolution 
about the major and minor axes). He determines the 
character of plane sections and the volume of seg- 
ments of such surfaces. Probably Archimedes also 
knew that these surfaces form the geometric locus of 
a point whose distances from a fixed point and a given 
plane are in a constant ratio. According to Proclus,* 
who is of importance as a commentator upon Euclid, 
the torus was also known — a surface generated by a 
circle of radius r revolving about an axis in its plane 
so that its center describes a circle of radius e. The 
cases r = ^, > ^, <ie were discussed. 

With methods of projection, also, the Greeks were 
not unacquainted, f Anaxagoras and Democritus are 
said to have known the laws of the. vanishing point 
and of reduction, at least for the simplest cases. Hip- 
parchus projects the celestial sphere from a pole upor 

* Major, Proklos Uber die Petita und Axtomata hti BtMid, 1875. 
t Wiener. 



the plane of the equator ; he is therefore the inventor 
of the stereographic projection which has come to be 
known by the name of Ptolemy. 



Among no other people of antiquity did geometry 
reach so high an eminence as among the Greeks. 
Their acquisitions in this domain were in part trans- 
planted to foreign soil, yet not so that (with the 
possible exception of arithmetic calculation) anything 
essentially new resulted. Frequently what was in- 
herited from the Greeks was not even fully under- 
stood, and therefore remained buried in the literature 
of the foreign nation. From the time of the Renais- 
sance, however, but especially from that of Descartes, 
an entirely new epoch with more powerful resources 
investigated the ancient treasures and laid them under 

Among the Romans independent investigation of 
mathematical truths almost wholly disappeared. What 
they obtained from the Greeks was made to serve 
practical ends exclusively. For this purpose parts of 
Euclid and Heron were translated. To simplify the 
work of the surveyors or agrimensores, important geo- 
metric theorems were collected into a larger work of 
which fragments are preserved in the Codex Arceri- 


anus. In the work of Vitruvius on architecture 
{_c. — 14) is found the value ir = 3^ which, though less 
accurate than Heron's value ir = 3^, was more easily 
employed in the duodecimal system.* Boethius has 
left a special treatise on geometry, but the contents 
are so paltry that it is safe to assume that he made 
use of an earlier imperfect treatment of Greek geom- 

Although the Hindu geometry is dependent upon 
the Greek, yet it has its own peculiarities due to the 
arithmetical modes of thought of the people. Certain 
parts of the Qulvasutras are geometric. These teach 
the rope-stretching already known to the Egyptians, 
i. e., they require the construction of a right angle by 
means of a rope divided by a knot into segments 16 
and 39 respectively, the ends being fastened to a seg- 
ment 36 (152-t- 36« = 39«). They also use the gnomon 
and deal with the transformation of figures and the 
application of the Pythagorean theorem to the multi- 
plication of a given square. Instead of the quadrature 
of the circle appears the circulature of the square, f 
i. e., the construction of a circle equal to a given 
square. Here the diameter is put equal to ^ of the 
diagonal of the square, whence follows ir = 3^ (the 
value used among the Romans). In other cases a 
process is carried on which yields the value ir = 3. 

The writings of Aryabhatta contain certain incor- 
rect formulae for the mensuration of the pyramid and 

*Cantor, I., p. 508. t Cantor, I., p. 601. 


sphere (for the pyramid V=z^B/i), but also a number 
of perfectly accurate geometric theorems. Aryabhatta 
gives the approximate value «• = j^^ = 3.1416. 
Brahmagupta teaches mensurational or Heronic ge- 
ometry and is familiar with the formula for the area 
of the triangle, 

and the formula for the area of the inscribed quadri- 

which he applies incorrectly to any quadrilateral. In 
his work besides w=Swe also find the value ir=T/10, 
but without any indication as to how it was obtained. 


Bhaskara likewise devotes himself only to algebraic 
geometry. For w he gives not only the Greek value 
^ and that of Aryabhatta f JfJJ, but also a value 
»=}f^ = 3.14166 ... Of geometric demonstrations 
Bhaskara knows nothing. He states the theorem, 
adds the figure and writes ''Behold ! "* 

In Bhaskara a transfer of geometry from Alexan- 
dria to India is undoubtedly demonstrable, and per- 
haps this influence extended still further eastward to 
the Chinese. In a Chinese work upon mathematics, 
composed perhaps several centuries after Christ, the 
Pythagorean theorem is applied to the triangle with 
sides 3, 4, 5; rope-stretching is indicated; the ver 

* Cantor, I., p. 614. 


tices of a figure are designated by letters after the 
Greek fashion ; rr is put equal to 3, and toward the 
end of the sixth century to ^/^ 

Greek geometry reached the Arabs in part directly 
and in part through the Hindus. The esteem, how- 
ever, in which the classic works of Greek origin were 
held could not make up for the lack of real produc- 
tive power, and so the Arabs did not succeed in a 
single point in carrying theoretic geometry, even in 
the subject of conic sections, beyond what had been 
reached in the golden age of Greek geometry. Only 
a few particulars may be mei>tioned. In Al Khowa- 
razmi is found a proof of the Pythagorean theorem 
consisting only of the separation of a square into 
eight isosceles right-angled triangles. On the whole 
Al Khowarazmi draws more from Greek than from 
Hindu sources. The classification of quadrilaterals 
is that of Euclid; the calculations are made after 
Heron's fashion. Besides the Greek value w=^ we 
find the Hindu values «-=|^f^ and «-=i/lO. Abul 
Wafa wrote a .book upon geometric constructions. 
In this are found combinations of several squares into 
a single one, as well as the construction of polyhedra 
after the methods of Pappus. After the Greek fash- 
ion the trisection of the angle occupied the attention 
of Tabit ibn Kurra, Al Kuhi, and Al Sagani. Among 
later mathematicians the custom of reducing a geo- 
metric problem to the solution of an equation is com- 
mon. It was thus that the Arabs by geometric solu- 


tions attained some excellent results, but results of no 
theoretic importance. 



Among the Western nations we find the first traces 
of geometry in the works of Gerbert, afterward known 
as Pope Sylvester II. Gerbert, as it seems, depends 
upon the Codex Arcerianus, but also mentions Pyth- 
agoras and Eratosthenes. * We find scarcely anything 
here besides field surveying as in Boethius. Some- 
thing more worthy first appears in Leonardo's (Fibo- 
nacci's) Practica geofnetriat\ of 1220, in which work 
reference is made to Euclid, Archimedes, Heron, and 
Ptolemy. The working over of the material handed 
down from the ancients, in Leonardo's book, is fairly 
independent. Thus the rectification of the circle 
shows where this mathematician, without making use 

of Archimedes, determines from the regular polygon 

of 96 sides the value «•= tfttt =3.1418. 


Since among the ancients no proper theory of star 
polygons can be established, it is not to be wondered 
at that the early Middle Ages have little to show in 
this direction. Star-polygons had first a mystic sig- 
nificance only ; they were used in the black art as the 
pentacle, and also in architecture and heraldry. Adel 

♦Cantor. T., p. 8io. tHankel, p. 344. 


ard of Bath went with more detail into the study of 
star-polygons in his commentary on Euclidean geom- 
etry ; the theory of these figures is first begun by Re- 

The first German mathematical work is the Deut 
sche Sphdra written in Middle High German by Conrad 
of Megenberg, probably in Vienna in the first half of 
the fourteenth century. The first popular introduc- 
tion to geometry appeared anonymously in the fif- 
teenth century, in six leaves of simple rules of con- 
struction for gecxnetric drawing. The beginning, con- 
taining the construction of BC perpendicular to AB 
by the aid of the right-angled triangle ABC in which 
BE bisects the hypotenuse A C, runs as follows : * 
''From geometry some useful bits which are written 
after this. 1. First to make a right angle quickly. 
Draw two lines across each other just about as you 
wish and where the lines cross each other there put 
an e. Then place the compasses with one foot upon 
the point <?, and open them out as far as you wish, 
and make upon each line a point. Let these be the 
letters a, ^, c^ all at one distance. Then make a line 
from ato b and from b to c. So you have a right angle 
of which here is an example.** 

This construction of a right angle, not given in 
Euclid but first in Proclus, appears about the year 
1500 to be in much more extensive use than the 
method of Euclid by the aid of the angle inscribed in 

* Giinther, p. 347. 



a semi- circle. By his knowledge of this last construc- 
tion Adam Riese is said to have humiliated an archi- 
tect who kaew how to draw a right angle only by the 
method of Proclus. 

Very old printed works on geometry in German are Dz Puech- 
len der fialen gerechtikait by Mathias Roriczer (1486) and Al- 
brecht DUrer's Underiveysung der messung mit dem zirckel 
wmf rwrA/s^Aey/ (Nuremberg, 1525). The former gives in rather 
unscientific manner rules for a special problem of Gothic architec- 
ture ; the latter, however, is a far more original work and on that 
account possesses more interest.* 

With the extension of geometric knowledge in 
Germany Widmann and Stifel were especially con- 
cerned. Widmann's geometry, like the elements of 
Euclid, begins with explanations : " Functus is a small 
thing that cannot be divided. Angulus is a comer 
which is made there by two lines, "f Quadrilaterals 
have Arab names, a striking evidence that the ancient 
Greek science was brought into the West by Arab in- 
fluence. Nevertheless, by Roman writers (Boethius) 
Widmann is led into many errors, as, e. g., when he 
gives the area of the isosceles triangle of side a as \c^. 

In Rudolff' s Coss, in the theory of powers, Stifel 
has occasion to speak of a subject which first receives 
proper estimation in the modern geometry, viz., the 
right to admit more than three dimensions. ** Since, 
however, we are in arithmetic where it is permitted 
to invent many things that otherwise have no form, 

♦ Gtinther in SchOfmiUh^t Zetttchrtft, XX., HI. 2. 

t Gerhardt. Geschichtt der Mathematik in DtuUchland, 1877. 


this also is permitted which geometry does not allow, 
namely to assume solid lines and surfaces and go be- 
yond the cube just as if there were more than three 
dimensions, which is, of course, against nature. . . . 
But we have such good indulgence on account of the 
charming and wonderful usage of Coss."* 

Stifel after the manner of Ptolemy extends the 
study of regular polygons and after the manner of 
Euclid the construction of regular solids. He dis- 
cusses the quadrature of the circle, considering the 
latter as a polygon of infinitely niany sides, and de- 
clares the quadrature impossible. According to Al- 
brecht Durer's Underweysung, etc., the quadrature of 
the circle is obtained when the diagonal of the square 
contains ten parts of which the diameter of the circle 
contains eight, i. e., 7r = 3|. It is expressly stated, 
however, that this is only an approximate construc- 
tion. **We should need to know quadraiura circuity 
that is the making equal a circle and a square, so that 
the one should contain as much as the other, but this 
has not yet been demonstrated mechanically by schol- 
ars ; but that is merely incidental ; therefore so that 
in practice it may fail only slightly, if at all, they may 
be made equal as follows, f 

*StifeI, Du Cnt Ckrisioffs Rudolffs. Mit schOnen Exempeln der Coss. 
Durch Michael Stifel Gebessert vnd sehr gemehrt. . . . Gegeben sum Haber- 
sten I bei KOnigsberg in Preussen | den letzten tag dess Herbstmonds | im 
Jar 1552. . . . Zn Amsterdam Getruckt bey Wilhem Janson. Im Jar 1615. 

tDfirer, Undervoeytung der messung mit dem zirckel vnd ricktwcheyt in 
Linien ebnen vnd gantten corporen. Durch Albrecht Diirer zusamen getxogn 
vnd su nutx alln kunstlieb habenden mit 2u gehOrigen fignren in truck 
gebracht im jar MDXXV. (Consists of vitr BUckUin,) 


Upon the mensuration of the circle* there appeared in 1584 a 
work by Simon van der Eycke in which the value 7r=: — — was 
given. By calculating the side of the regular polygon of 192 sides 
Ludolph van Ceulen found (probably in 1585) that tt <8.14205 < 
— — . In his reply Simon v. d, Eycke determined tts* 8. 1446055, 
whereupon L. v. Ceulen in 1586 computed tr between 8.142732 
and 8.14103. Ludolph van Ceulen's papers contain a value of tt 
to 85 places, and this value of the Ludolphian number was put 
upon his tombstone (no longer known) in St. Peter's Church in 
Leyden. Ceulen's investigations. led Snellius, Huygens, and others 
to further studies. By the theory of rapidly converging series it 
was first made possible to compute tt to 500 and more decimals, f 

A revival of geometry accompanied the activity of 
Vieta and Kepler. With these investigators begins a 
period in which the mathematical spirit commences 
to reach out beyond the works of the ancients. J Vieta 
completes the analytic method of Plato ; in an ingeni 
ous way he discusses the geometric construction of 
roots of equations of the second and third degrees ; 
he also solves in an elementary manner the problem 
of the circle tangent to three given circles. Still 
more important results are secured by Kepler. For 
him geometry furnishes the key to the secrets of the 
world. With sure step he follows the path of induc- 
tion and in his geometric investigations freely con- 
forms to Euclid. Kepler established the symbolism 
of the '* golden section," that problem of Eudoxus 

♦Rudio, F., Das Problem von der Quadratur des Zirkels^ Ziirich, 189a 
tD. Bierens de Haan in Nieuw. Arch., I., Cantor, II., p. 551. 
X Chasles. 


Stated in the sixth book of Kuclid' s E/emenfs \ **To 
divide a limited straight line in extreme and mean 
ratio."* This pi-oblem, for which Kepler introduced 
the designation sectio divina as well as proportio divina^ 
is in his eyes of so great importance that he expresses 
himself: ** Geometry has two great treasures: one is 
the theorem of Pythagoras, the other the division of 
a line in extreme and mean ratio. The first we may 
compare to a mass of gold, the second we may call a 
precious jewel." 

The expression " golden section " is of more modem origin. 
It occurs in none of the text books of the eighteenth century and 
appears to have been formed by a transfer from ordinary arithme- 
tic. In the arithmetic of the sixteenth and seventeenth centuries 
the rule of three is frequently called the "golden rule." Since the 
beginning of the nineteenth century this golden rule has given way 
more and more before the so-called Schlussrechnen (analysis) of 
the Pestalozzi school. Consequently in place of the "golden rule," 
which is no longer known to the arithmetics, there appeared in the 
elementary geometries about the middle of the nineteenth century 
the "golden section," probably in connection with contemporary 
endeavors to attribute to this geometric construction the impor- 
tance of a natural law. 

Led on by his astronomical speculations, Kepler 
made a special study of regular polygons and star- 
polygons. He considered groups of regular polygons 
capable of elementary construction, viz. , the series of 
polyg:ons with the number of sides given by 4 •2", 
3 •2", 5- 2", 15- 2" (from « = on), and remarked that 

*Sonnenbarg, Der goldene Scknitt^ i88z. 


a regular heptagon cannot be constructed by the help 
of the straight line and circle alone. Further there is 
no doubt that Kepler well understood the Conies of 
Apollonius and had experience in the solution of prob- 
lems by the aid of these curves. In his works we 
first find the term **foci** for those points of conic 
sections which in earlier usage are known as puncia 
ex coMparatione, puncta ex applicatione facta, umbilici^ 
or ** poles";* also the term *ieccentricity" for the 
distance from a focus to the center divided by the 
semi- major axis, of the curve of the second order, and 
the name "eccentric anomaly" for the angle Z*'^^, 
where OA is the semi-major axis of an ellipse and /^ 
the point in which the ordinate of a point P on the 
curve intersects the circle upon the major axis.f 

Also in stereometric investigations, which had been 
cultivated to a decided extent by Dtirer and Stifel, 
Kepler is preeminent among his contemporaries. In 
his Harmonice Mundi he deals not simply with the 
five regular Platonic and thirteen semi-regular Archi- 
medean solids, but also with star-polygons and star- 
dodecahedra of twelve and twenty vertices. Besides 
this we find the determination of the volumes of solids 
obtained by the revolution of conies about diameters, 
tangents, or secants. Similar determinations of vol- 
umes were effected by Cavalieri and Guldin. The 
former employed a happy modification of the method 

♦ C. Taylor, in Cambr. Proe., IV. 

t Baltzer, R., Analytische Geometrie, x88a. 


of exhaustions, the latter used a rule already known 
to Pappus but not accurately established by him. 

To this period belong the oldest known attempts 
to solve geometric problems with only one opening of 
the compasses, an endeavor which first found accurate 
scientific expression in Steiner's Geometrische Con- 
struktioneny ausgefuhrt tnittels der geraden Linie und 
eines festtn Kreises (1833). The first traces of such 
constructions go back to Abul Wafa.* From the Arabs 
they were transmitted to the Italian school where they 
appear in the works of Leonardo da Vinci and Cardan. 
The latter received his impulse from Tartaglia who 
used processes of this sort in his problem-duel with 
Cardan and Ferrari. They also occur in the Resolutio 
omnium Euclidis probUmatum (Venice, 1553) of Bene- 
dictis, a pupil of Cardan, in the Geotnetria deuisch and 
in the construction of a regular pentagon by Durer. 
In his Underweysung, etc., Diirer gives a geometrically 
accurate construction of the regular pentagon but also 
an approximate construction of the same figure to be 
made with a circle of fixed radius. 

This method of conttrncting a regular pentagon on AB is as 
follows : About A and B as centers, with radius AB, construct cir- 
cles intersecting in C and D. The circle about Z> as a center with 
the same radius cuts the circles with centers at A and ^ in ^ and 
F and the common chord CD in G. The same circles are cut by 
FG and EG in J and H. A J and BH are sides of the regular 
pentagon. (The calculation of this symmetric pentagon shows 

♦Gflnther in SchWmilch's Zeitschrift, XX. Cantor. I., p. 700. 


ffBA T^IOS^W. whiU the coicespondiag angle of the reRular 
pentagon if 1U8°.) 

In DDrer and alt bis successors who write upon rules at geo- 
rnelric cooslruclion, we find an approximate construction of the 
regular heptagon ; ' ' The side of the regular heptagon is half that 
of the equilateral triangle," while from calculation the half side 
of the equilateral triangle^D.9Q6 of the side of the heptagon 
Daniel Schweoler likewise gave constructions with a single opening 
of the compasses in his Geomelria fractica nova et aucta (1625). 
Dllrer. as is manifest from bis work Undenveysung dermessung. 
etc., already cited several times, also rendered decided service in 
the theory of higher cnrves. He gave a general conception of the 
notion of asymptotes and fonnd as new forms of higher carves cei 
lain cyclic cnrves and mussel-shaped lines 

From the fifteenth century on, the methods of pro- 
jection make a further advance. Jan van Eyck* in 
the great altar painting in Ghent makes use of the 
laws of perspective, e. g., in the application of the 


vanishing point, but without a mathematical grasp of 
these laws. This is first accomplished by Albrecht 
Durer who in his Underweysung der messung mit dem 
zirckel und richischeyt makes use of the point of sight 
and distance-point and shows how to construct the 
perspective picture from the ground plan and eleva- 
tion. In Italy perspective was developed by the archi- 
tect Brunelleschi and the sculptor Donatello. The 
first work upon this new theory is due to the architect 
Leo Battista Alberti. In this he explains the perspec- 
tive image as the intersection of the pyramid of visual 
rays with the picture-plane. He also mentions an in- 
strument for constructing it, which consists of a frame 
with a quadratic net- work of threads and a similar 
net-work of lines upon the drawing surface. He also 
gives the method of the distance-point as invented by 
him, by means of which he then pictures the ground 
divided into quadratic figures.* This process received 
a further extension at the hands of Piero della Fran- 
cesca who employed the vanishing points of arbitrary 
horizontal lines. 

In German territory perspective was cultivated 
with special zeal in Nuremberg where the goldsmith 
Lencker, some decades after Durer, extended the lat- 
ter's methods. The first French study of perspective 
is due to the artist J. Cousin (1560) who in his Livre 
de la perspective made use of the point of sight and the 
distance-point, besides the vanishing points of hori- 

♦ wiener. 


zontal lines, after the manner of Piero. Guido Ubaldi 
goes noticeably further when he introduc'es the van- 
ishing point of series of parallel lines of arbitrary di- 
rection. What Ubaldi simply foreshadows, Simon 
Stevin clearly grasps in its principal features, and in 
an important theorem he lays the foundation for the 
development of the theory of coUineation. 



Since the time of Apollonius many centuries had 
elapsed and yet no one had succeeded in reaching the 
full height of Greek geometry. This was partly be- 
cause the sources of information were relatively few, 
and attainable indirectly and with difficulty, and partly 
because men, unfamiliar with Greek methods of in- 
vestigation, looked upon them with devout astonish- 
ment. From this condition of partial paralysis, and 
of helpless endeavor longing for relief, geometry was 
delivered by Descartes. This was not by a simple ad- 
dition of related ideas to the old geometry, but merely 
by the union of algebra with geometry, thus giving 
rise to analytic geometry. 

By way of preparation many mathematicians, first 
of all Apollonius, had referred the most important ele- 
mentary curves, namely the conies, to their diameters 
and tangents and had expressed this relation by equa- 


tions of the first degree between areas, so that cer- 
tain relations were obtained between line-segments 
identical with abscissas and ordinates. 

In the conies of Apollonius we find expressions 
which have been translated **ordinatim applicatae" 
and "abscissae." For the former expression Fermat 
used "applicate" while others wrote "ordinate." 
Since the time of Leibnitz (1692) abscissas and ordi- 
nates have been called "co-ordinates."* 

Even in the fourteenth century we find as an ob- 
ject of study in the universities a kind of co-ordinate 
geometry, the "latitudines formarum." " Latitude" f 
signified the ordinate, "longitudo" the abscissa of a 
variable point referred to a system of 'rectangular co- 
ordinates, and the different positions of this point 
formed the "figura." The technical words longitude 
and latitude had evidently been borrowed from the 
language of astronomy. In practice of this art Oresme 
confined himself to the first quadrant in which he 
dealt with straight lines, circles, and even the para- 
bola, but always so that only a positive value of a co- 
ordinate was considered. 

Among the predecessors of Descartes we reckon, 
besides Apollonius, especially, Vieta, Oresme, Cava- 
lieri, Roberval, and Fermat, the last the most distin- 
guished in this field ; but nowhere, even by Fermat, 
had any attempt been made to refer several curves of 

*Balt2er, R., Anafyttscke Geomttrie, 1883. 
tGiinther, p. i8z. 


different orders simultaneously to one system of co- 
ordinates, which at most possessed special significance 
for one of the curves. It is exactly this thing which 
Descartes systematically accomplished. 

The thought with which Descartes made the lawt* 
of arithmetic subservient to geometry is set forth by 
himself in the following manner : * 

"All problems of geometry may be reduced to such 
terms that for their construction we need only to know 
the length of certain right lines. And just as arith- 
metic as a whole comprises only four or five opera- 
tions, viz., addition, subtraction, multiplication, divi- 
sion, and evolution, which may be considered as a 
kind of division, so in geometry to prepare the lines 
sought to be known we have only to add other lines 
to them or subtract others from them ; or, having one 
which I call unity (so as better to refer it to numbers), 
which can ordinarily be taken at pleasure, having two 
others to find a fourth which shall be to one of these 
as the other is to unity, which is the same as multi 
plication ;f or to find a fourth which shall be to one 
of the two as unity is to the other which is the same 
as division ; J or finally to find one or two or several 
mean proportionals between unity and any other line, 
which is the same as to extract the square, cube, . . . 
root.§ I shall not hesitate to introduce these terms 

* Marie, M., Histoire des Sciences Mathiinatique* et Physiqut*^ 1883-1887. 

Vc'.a — b'.i^ c = ab. 

Xc:a = i:b, c = a:b. 

%i : a — a '. b = b : c — c '. d ^ . . 

. gives a=yb = fc = ^d.. 


of arithmetic into geometry in order to render myself 
more intelligible. It should be observed that, by a*, 
^, and similar quantities, I understand as usual sim- 
ple lines, and that I call them square or cube only so 
as to employ the ordinary terms of algebra." (a* is 
the third proportional to unity and a, or 1 : a = <i:a*, 
and similarly b\b^=:^l^\I^,^ 

This method of considering arithmetical expres- 
sions was especially influenced by the geometric dis- 
coveries of Descartes. As Apollonius had already de- 
termined points of a conic section by parallel chords, 
together with the distances from a tangent belonging 
to the same system, measured in the direction of the 
conjugate diameter, so with Descartes every point of 
a curve is the intersection of two straight lines. Apol- 
lonius and all his successors, however, apply such 
systems of parallel lines only occasionally and that for 
the sole purpose of presenting some definite property 
of the conies with especial distinctness. Descartes, 
on the contrary, separates these systems of parallel 
lines from the curves, assigns them an independent 
existence and so obtains for every point on the curve 
a relation between two segments of given direction, 
which is nothing else than an equation. The geo- 
metric study of the properties of this curve can then 
be replaced by the discussion of the equation after the 
methods of algebra. The fundamental elements for 
the determination of a point of a curve are its co-or- 
dinates, and from long known theorems it was evident 


that a point of the plane can be fixed by two co-ordi 
nates, a point of space by three. 

Descartes's Geometry is not, perhaps, a treatise 
on analytic geometry, but only a brief sketch which 
sets forth the foundations of this theory in outline. 
Of the three books which constitute the whole work 
only the first two deal with geometry ; the third is of 
algebraic nature and contains the celebrated rule of 
signs illustrated by a simple example, as well as the 
solution of equations of the third and fourth degrees 
with the construction of their roots by the use of 

The first impulse to his geometric reflections was 
due, as Descartes himself says, to a problem which 
according to Pappus had already occupied the atten- 
tion of Euclid and ApoUonius. It is the problem to 
find a certain locus related to three, four, or several 
lines. Denoting the distances, measured in given di- 
rections, of a point P from the straight lines ^1, ^j . . . 
gn by ^\y e%. . . e^y respectively, we shall have 

for three straight lines : — i— ?- = ky 


e\ e% 
for four straight lines : = k, 

for five straight lines : ^ * ' = k, 

a ei e^ 

and so on. The Greeks originated the solution of the 
first two cases, which furnish conic sections. No ex- 
ample could have shown better the advantage of the 


new method. For the case of three lines Descartes 
denotes a distance by y^ the segment of the corres- 
ponding line between the foot of this perpendicular 
and a fixed point by x^ and shows that every other 
segment involved in the problem can be easily con- 
structed. Further he states "that if we allow y to 
grow gradually by infinitesimal increments, x will 
grow in the same way and thus we may get infinitely 
many points of the locus in question." 

The curves with which Descartes makes us gradu- 
ally familiar he classifies so that lines of the first and 
second orders form a first group, those of the third 
and fourth orders a second, those of the fifth and 
sixth orders a third, and so on. Newton was the first 
to call a curve, which is defined by an algebraic equa- 
tion of the «th degree between parallel co-ordinates, a 
line of the «th order, or a curve of the (« — l)th class. 
The division into algebraic and transcendental curves 
was introduced by Leibnitz; previously, after the 
Greek fashion, the former had been called geometric, 
the latter mechanical lines.* 

Among the applications wfiich Descartes makes^ 
the problem of tangents is prominent. This he treats 
in a peculiar way : Having drawn a normal to a curve 
at the point Py he describes a circle through P with 
the center at the intersection of this normal with the 

*Balt£er, R., AnafyttscAg Gtometrie, 1883. Up to the time of Descartes 
all lines except straight lines and conies were called mechanical. He was 
the first to apply the term geometric lines to curves of degree hlghar than 
the second. 


X-axis, and asserts that this circle cuts the curve at P 
in. two consecutive points; i. e., he states the condi- 
tion that after the elimination of x the equation in y 
shall have a double root. 

A natural consequence of the acceptance of the 
Cartesian co-ordinate system was the admission of 
negative roots of algebraic equations. These negative 
roots had now a real significance ; they could be rep- 
resented, and hence were entitled to the same rights 
as positive roots. 

In the period immediately following Descartes, 
geometry was enriched by the labors of Cavalieri, 
Fermat, Roberval, Wallis, Pascal, and Newton, not 
at first by a simple application of the co-ordinate ge- 
ometry, but often after the manner of the ancient 
Greek geometry, though with some of the methods 
essentially improved. The latter is especially true of 
Cavalieri, the inventor of the method of indivisibles,* 
which a little later was displaced by the integral ca 
cuius, but may find a place here since it rendered ser- 
vice to geometry exclusively. Cavalieri enjoyed work- 
ing with the geometry of the ancients. For example, 
he was the first to give a satisfactory proof of the so- 
called Guldin's rule already stated by Pappus. His 
chief endeavor was to find a general process for the 
determination of areas and volumes as well as centers 
of gravity, and for this purpose he remodelled the 

* In French works MHhode des indivisibles^ originally in the work Geo- 
metria indivisibilibus continuorum nova quadam ratiou* ^ramotm^ Bologna, 



method of exhaustions. Inasmuch as Cavalieri's 
method, of which he was master as early as 1629, may 
even to-day replace to advantage ordinary integration 
In elementary cases, its essential character may be set 
forth in brief outline.* 

If y=/(^x) is the equation of a curve in rectangu- 
lar co-ordinates, and he wishes to determine the area 
bounded by the axis of Xy a portion of the curve, and 
the ordinates corresponding to xq and Xu Cavalieri 
divides the difference xi — Xq into n equal parts. Let 
^ represent such a part and let n be taken very large. 
An element of the surface is then z^Ay = /i/{x), and 

the whole surface becomes 



For « = 00 we eviden tly get exactly 



But this is not the quantity which Cavalieri seeks to 
determine. He forms only the ratios of portions of 
the area sought, to the rectangle with base xi — ^o 
and altitude yi, so that the quantity to be determined 
is the following : 

«— 1 n—l 

^/i'/ (xo + n/i) ^^(^0 + nA) 

Cavalieri applies this formula, which he derives in 

* Marie. 


complete generality from grounds of analogy, only to 
the case where /(jtr) is of the form Ao^ (jw:=2, 3, 4). 
The extension to further cases was made by Rober- 
val, Wallis, and Pascal. 

In the simplest cases the method of indivisibles gives the fol- 
lowing results.* For a parallelogram the indivisible quantity or 
element of surface is a parallel to the base; the number of indi- 
visible quantities is proportional to the altitude ; hence we have 
as the measure of the area of the parallelogram the product of the 
measures of the base and altitude. The corresponding conclusion 
holds for the prism. In order to compare the area of a triangle 
with that of the parallelogram of the same base and altitude, we 
decompose each into elements by equidistant parallels to the base. 
The elements of the triangle are then, beginning with the least, 1, 
2, 8, ...» ; those of the parallelogram, it, it, ... it Hence the 

Triangle ^ 1 +2+ ■ ■ +» ^ i»(» + l) ^ ^fl ■ AV 
Parallelogram «•« «' 2\«/' 

whence for 11 ^ 00 we get the value \. For the corresponding solids 
we get likewise 

Pyramid _ 1« +2» -f . . . + »« _ tif (it + 1) (2if + 1) 
Prism «■ 11^ 


After the lapse of a few decades this analytic- 
geometric method of Cavalieri's was forced into the 
background bj' the integral calculus, which could be 
directly applied in all cases. At first, however, Rober- 
val, known by his method of tangents, trod in the 
footsteps of Cavalieri. Wallis used the works of Des- 

* Marie. 


cartes and Cavalieri simultaneously, and considered 
especially curves whose equations were of the form 
y^=j^j m integral or fractional, positive or negative. 
His chief service consists in this, that in his brilliant 
work he put a proper estimate upon Descartes's dis- 
covery and rendered it more accessible. In this work 
Wallis also defines the conies as curves of the second 
degree, a thing never before done in this definite 

Pascal proved to be a talented disciple of Cavalieri 
and Desargues. In his work on conies, composed 
about 1639 but now lost (save for a fragment),* we 
find Pascal's theorem of the inscribed hexagon or 
Hexagramma mysticum as he termed it, which Bessel 
rediscovered in 1820 without being aware of Pascal's 
earlier work,f also the theorem due to Desargues that 
if a straight line cuts a conic in P and Q^ and the 
sides of an inscribed quadrilateral in Ay B^ C, />, we 
have the following equation : 


Pascal's last work deals with a curve called by him 
the roulette, by Roberval the trochoid, and generally 
known later as the cycloid. Bouvelles (1503) already 
knew the construction of this curve, as did Cardinal 
von Cusa in the preceding century. \ Galileo, as is 
shown by a letter to Torricelli in 1639, had made (be 

* Cantor, II., p. 6a2. t Bianco in Torino Att.^ XXI. 

^Giinther has shown the last statement to be incorrect. 


ginning in 1590) an exhaustive study of rolling curves 
in connection with the construction of bridge arches. 
The qukdrature of the cycloid and the determination 
of the volume obtained by revolution about its axis 
had been effected by Roberval, and the construction 
of the tangent by Descartes. In the year 1658 Pascal 
was able to determine the length of an arc of a cy- 
cloidal segment, the center of gravity of this surface, 
and the corresponding solid of revolution. Later the 
cycloid appears in physics as the brachistochrone and 
tautochrone, since it permits a body sliding upon it to 
pass from one fixed point to another in the shortest 
time, while it brings a material ppint oscillating upon 
it to its lowest position always in the same time. 
Jacob and John Bernoulli, among others, gave atten- 
tion to isoperimetric problems ; but only the former 
secured any results of value, by furnishing a rigid 
method for their solution which received merely an 
unimportant simplification from John Bernoulli. (See 
pages 178-179.) 

The decades following Pascal's activity were in 
large part devoted to the study of tangent problems 
and the allied normal problems, but at the same time 
the general theory of plane curves was constantly 
developing. Barrow gave a new method of determin- 
ing tangents, and Huygens studied evolutes of curves 
and indicated the way of determining radii of curva- 
ture. From the consideration of caustics, Tschirn- 
hausen was led to involutes and Maclaurin constructed 


the Circle of curvature at any point of an algebraic 
curve. The most important extension of this theory 
was made in Newton's Enumeratio linearum tertii or- 
dints (1706). This treatise establishes the distinction 
between algebraic and transcendental curves. It then 
makes an exhaustive study of the equation of a curve 
of the third order, and thus finds numerous such curves 
which may be represented as *' shadows" of five types, 
a result which involves an analytic theory of perspec- 
tive. Newton knew how to construct conies from five 
tangents. He came upon this discovery in his en- 
deavor to investigate ''after the manner of the an- 
cients" without analytic geometry. Further he con- 
sidered multiple points of a curve at a finite distance 
and at infinity, and gave rules for investigating the 
course of a curve in the neighborhood of one of its 
points (** Newton's parallelogram" or ** analytic tri- 
angle "), as also for the determination of the order of 
contact of two curves at one of their common points. 
(Leibnitz and Jacob Bernoulli had also written upon 
osculations ; Plucker (1831) called the situation where 
two curves have k consecutive points in common **a 
^-pointic contact"; in the same case Lagrange (1779) 
had spoken of a ** contact of (k — l)th order. ")t 

Additional work was done by Newton's disciples. 
Cotes and Maclaurin, as well as by Waring. Mac- 
laurin made interesting investigations upon corre- 

* Baltzer. 

tCayley, A., Address to the British Association^ etc., 1883. 


spending points of a curve of the third order, and 
thus showed that the theory of these curves was much 
more comprehensive than that of conies. Euler like- 
wise entered upon these investigations in his paper 
Sur une contradiction apparente dans la thSorie des courbcs 
planes (Berlin, 1748), where it is shown that by eight in- 
tersections of two curves of the third order the ninth is 
completely determined. This theorem, which includes 
Pascal's theorem for conies, introduced point groups, 
or systems of points of intersection of two curves, into 
geometry. This theorem of Euler's was noticed in 
1750 by Cramer who gave special attention to the sin- 
gularities of curves in his works upon the intersection 
of two algebraic curves of higher order; hence the 
obvious contradiction between the number of points 
determining a plane curve and the number of inde- 
pendent intersections of two curves of the same order 
bears the name of ** Cramer's paradox." This contra- 
diction was solved by Lam6 in 1818 by the principle 
which bears his name.* Partly in connection with 
known results of the Greek geometry, and partly in- 
dependently, the properties of certain algebraic and 
transcendental curves were investigated. A curve 
which is formed like the conchoid of Nicomedes, if 
we replace the straight line by a circle, is called by 

•Loria, G., Dit haupttachlichsten Theorien der Geometrie in threrfrUhe- 
ren undjetnigen Entwicklung. Deutsch von Scbiitte, 1888. For a more accu- 
rate acconnt of Cramer's paradox, in which proper credit is given to Mac- 
laurin's discovery, see Scott, C. A., " On the Intersections of Plane Curves " 
Bull. Am Math. Soc, March, 1898. 


Roberval the lima9on of Pascal. The cardioid of the 
eighteenth century is a special case of this spiral. If, 
with reference to two fixed points A^ By a point P 
satisfies the condition that a linear function of the 
distances PA^ PB has a constant value, then is the 
locus oi P Si Cartesian oval. This curve was found by 
Descartes in his studies in dioptrics. For PA • PB = 
constant, we have Cassini's oval, which the astronomer 
of Louis XIV. wished to regard as the orbit of a planet 
instead of Kepler's ellipse. In special cases Cassini's 
oval contains a loop, and this form received from 
Jacob Bernoulli (1694) the name lemniscate. With 
the investigation of the logarithmic curve yr=a^ wsls 
connected the study made by Jacob and John Ber- 
noulli, Leibnitz, Huygens, and others, of the curve of 
equilibrium of an inextensible, flexible thread. This 
furnished the catenary {catenariay 1691), the idea of 
which had already occurred to Galileo.* The group 
of spirals found by Archimedes was enlarged in the 
seventeenth and eighteenth centuries by the addition 
of the hyperbolic, parabolic, and logarithmic spirals, 
and Cotes's lituus (1722). In 1687 Tschirnhausen de- 
fined a quadra trix, differing from that of the Greeks, 
as the locus of a point /', lying at the same time upon 
LQ\\BO and upon MP\\OA {OAB is a quadrant), 
where L moves over the quadrant and M over the 
radius OB uniformly. Whole systems of curves and 
surfaces were considered. Here belong the investiga- 

« Cantor, III., p. 211. 


tions of involutes and evolutes, envelopes in general, 
due to Huygens, Tschirnhausen, John Bernoulli, 
Leibnitz, and others. The consideration of the pen- 
cil of rays through a point in the plane, and of the 
pencil of planes through a straight line in space, was 
introduced by Desargues, 1639.* 

The extension of the Cartesian co-ordinate method 
to space of three dimensions was effected by the labors 
of Van Schocten, Parent, and Clairaut.f Parent rep- 
resented a surface by an equation involving the three 
co-ordinates of a point in space, and Clairaut per- 
fected this new procedure in a most essential manner 
by a classic work upon curves of double curvature. 
About seventeen years later Euler established the ana- 
lytic theory of the curvature of surfaces, and the clas- 
sification of surfaces in accordance with theorems 
analogous to those used in plane geometry. He gives 
formulae of transformation of space co-ordinates and 
a discussion of the general equation of surfaces of the 
second order, with their classification. Instead of 
Euler's names: **elliptoid, elliptic-hyperbolic, hyper- 
bolic-hyperbolic, elliptic-parabolic, parabolic- hyper- 
bolic surface," the terms now in use, "ellipsoid, hyper- 
boloid, paraboloid," were naturalized by Biot and 
Lacroix. J 

Certain special investigations are worthy of men- 
tion. In 1663 Wallis studied plane sections and 
effected the cubature, of a conoid with horizontal di- 

*Baltzer. tLoria. ^Baltzer. 


recting plane whose generatrix intersects a vertical 
directing straight line and vertical directing circle 
{cono-cuneus). To Wren we owe an investigation of 
the hyperboloid of revolution of two sheets (1669) 
which he called **cylindroid.'* The domain of gauche 
curves, of which the Greeks knew the helical curve 
of Archytas and the spherical spiral corresponding in 
formation to the plane spiral of Archimedes, found an 
extension in the line which cuts under a constant an- 
gle the meridians of a sphere. Nunez (1546) had 
recognized this curve as not plane, and Snellius (1624) 
had given it the name loxodromia sphaerica. The prob- 
lem of the shortest line between two points of a sur- 
face, leading to gauche curves which the nineteenth 
century has termed ''geodetic lines," was stated by 
John Bernoulli (1698) and taken in hand by him with 
good results. In a work of Pitot in 1724 (printed in 
1726)* upon the helix, we find for the first time the 
expression ligne it double courbure, line of double curva- 
ture, for a gauche curve. In 1776 and 1780 Meusnier 
gave theorems upon the tangent planes to ruled sur- 
faces, and upon the curvature of a surface at one of 
its points, as a preparation for the powerful develop- 
ment of the theory of surfaces soon to begin, f 

There are still some minor investigations belong- 
ing to this period deserving of mention. The alge- 
braic expression for the distance between the centers 
of the inscribed and circumscribed circles of a triangle 

•Cantor, III., p. 428. t Baltzer. 


was determined by William Chappie (about 1746), 
afterwards by Landen (1755) and Euler (1765).* In 
1769 Meister calculated the areas of polygons whose 
sides, limited by every two consecutive vertices, inter- 
sect so that the perimeter contains a certain number 
of double points and the polygon breaks up into cells 
with simple or multiple positive or negative areas. 
Upon the areas of such singular polygons MObius pub- 
lished later investigations (1827 and 1865).* Saurin 
considered the tangents of a curve at multiple points 
and Ceva starting from static theorems studied the 
transversals of geometric figures. Stewart still further 
extended the theorems of Ceva, while Cotes deter- 
mined the harmonic mean between the segments of a 
secant to a curve of the nth order reckoned from a 
fixed point. Carnot also extended the theory of trans- 
versals. Lhuilier solved the problem : In a circle to 
inscribe a polygon of n sides passing through n fixed 
points. Brianchon gave the theorem concerning the 
hexagon circumscribed about a conic dualistically re- 
lated to Pascal's theorem upon the inscribed hexagon. 
The application of these two theorems to the surface 
of the sphere was effected by Hesse and Thieme. In 
the work of Hesse a Pascal hexagon is formed upon 
the sphere by six points which lie upon the intersec- 
tion of the sphere with a cone of the second order 
having its vertex at the center of the sphere. Thieme 
selects a right circular cone. The material usually 

* Fortschrttte^ 1887, p. 32. fBaltser. 


taken for the elementary geometry of the schools has 
among other things received an extension through 
numerous theorems upon the circle named after K. 
W. Feuerbach (1822), upon symmedian lines of a 
triangle, upon the Grebe point and the Brocard fig- 
ures (discovered in part by Crelle, 1816 ; again intro- 
duced by Brocard, 1875).* 

The theory of regular geometric . figures received 
its most important extension at the hands of Gauss, 
who discovered noteworthy theorems upon the possi- 
bility or impossibility of elementary constructions of 
regular polygons. (See p. 160.) Poinsot elaborated 
the theory of the regular polyhedra by publishing his 
views on the five Platonic bodies and especially upon 
the *< Kepler- Poinsot regular solids of higher class, " 
viz., the four star-polyhedra which are formed from 
the icosahedron and dodecahedron. These studies 
were continued by Wiener, Hessel, and Hess, with 
the removal of certain restrictions, so that a whole 
series of solids, which in an extended sense may be 
regarded as regular, may be added to those named 
above. Corresponding studies for four-dimensional 
space have been undertaken by Scheffler, Rudel, 
Stringham, Hoppe, and Schlegel. They have deter- 
mined that in such a space there exist six regular fig- 
ures of which the simplest has as its boundary five 
tetrahedra. The boundaries of the remaining five fig- 

♦Lieber, Ueber die Gegenmittellinie, den Grebe* schen Punkt und den Bro- 
card' schen Kreis, 1886- J 888. 


ures require 16 or 600 tetrahedra, 8 hexahedra, 24 
octahedra, 120 dodecahedra. It may be mentioned 
further that in 1849 the prismatoid was introduced 
into stereometry by E. F. August, and that Schubert 
and Stoll so generalised the Apollonian contact prob- 
lem as to be able to give the construction of the six- 
teen spheres tangent to four given spheres. 

Projective geometry, called less precisely modern 
geometry or geometry of position, is essentially a 
creation of the nineteenth century. The analytic ge- 
ometry of Descartes, in connection with the higher 
analysis created by Leibnitz and Newton, had regis- 
tered a series of important discoveries in the domain 
of the geometry of space, but it had not succeeded in 
obtaining a satisfactory proof for theorems of pure 
geometry. Relations of a specific geometric character 
had, however, been discovered in constructive draw- 
ing. Newton's establishment of his five principal 
types of curves of the third order, of which the sixty- 
four remaining types may be regarded as projections, 
had also given an impulse in the same direction. Still 
more important were the preliminary works of Carnot, 
which paved the way for the development of the new 
theory by Poncelet, Chasles, Steiner, and von Staudt. 
They it was who discovered '*the overflowing spring 
of deep and elegant theorems which with astonishing 
facility united into an organic whole, into the graceful 
edifice of projective geometry, which, especially with 


reference to the theory of curves of the second order, 
may be regarded as the ideal of a scientific organism."* 

Projective geometry found its earliest unfolding on 
French soil in the Giotnetrie descriptive of Monge whose 
astonishing power of imagination, supported by the 
methods of descriptive geometry, discovered a host of 
properties of surfaces and curves applicable to the 
classification of figures in space. His work created 
"for geometry the hitherto unknown idea of geomet- 
ric generality and geometric elegance, "f and the im- 
portance of his works is fundamental not only for the 
theory of projectivity but also for the theory of the 
curvature of surfaces. To the introduction of the 
imaginary into the considerations of pure geometry 
Monge likewise gave the first impulse, while his pupil 
Gaultier extended these investigations by defining the 
radical axis of two circles as a secant of the same 
passing through their intersections, whether real or 

The results of Monge's school thus derived, which 
were more closely related to pure geometry than to 
the analytic geometry of Descartes, consisted chiefly 
in a series of new and interesting theorems upon sur- 
faces of the second order, and thus belonged to the 
same field that had been entered upon before Monge's 
time by Wren (1669), Parent and Euler. That Monge 

♦Brill, A.. Antrittsrede in TUingen, 1884. 

+ Hankel, Dit EUmenti der profekttvuchen Geometrie in tynthetischtr Be- 
hantUnng, 1875. 


did not hold analytic methods in light esteem is shown 
by his Application de Valgebre d la giomitrie (1805) in 
which, as Plucker says, "he introduced the equation 
of the straight line into analytic geometry, thus laying 
the foundation for the banishment of all constructions 
from it, and gave it that new form which rendered 
further extension possible. " 

While Monge was working by preference in the 
space of three dimensions, Carnot was making a spe- 
cial study of ratios of magnitudes in figures cut by 
transversals, and thus, by the introduction of the nega- 
tive, was laying the foundation for a gSom^trie de posi- 
tion which, however, is not identical with the Geometrie 
der Lage of to-day. Not the most important, but the 
most noteworthy contribution for elementary school 
geometry is that of Carnot*s upon the complete quadri- 
lateral and quadrangle. 

Monge and Carnot having removed the obstacles 
which stood in the way of a natural development of 
geometry upon its own territory, these new -ideas could 
now be certain of a rapid developinent in well-pre- 
pared soil. Poncelet furnished the seed. His work. 
Traits des propriMs projectives des figures^ which ap- 
peared in 1822, investigates those properties of figures 
which remain unchanged in projection, i. e., their in- 
variant properties. The projection is not made here, 
as in Monge, by parallel rays in a given direction, but 
by central projection, and so after the manner of per- 
spective. In this way Poncelet came to introduce 


the axis of perspective and center of perspective (ac- 
cording to ChasleSy axis and center of homology) in 
the consideration of plane figures for which Desargues 
had already established the fundamental theorems. 
In 1811 Servois had used the expression **pole of a 
straight line,*' and in 1813 Gergonne the terms "polar 
of a point" and ''duality," but in 1818 Poncelet de- 
veloped some observations made by Lahire in 1685, 
upon the mutual correspondence of pole and polar in 
the case of conies, into a method of transforming fig- 
ures into their reciprocal polars. Gergonne recog- 
nized in this theory of reciprocal polars a principle 
whose beginnings were known to Vieta, Lansberg, 
and Snellius, from spherical geometry. He called it 
the "principle of duality" (1826). In 1827 Gergonne 
associated dualistically with the notion of order of a 
plane curve that of its class. The line is of the «th 
order when a straight line of the plane cuts it in n 
points, of the nth class when from a point in the plane 
n tangents can be drawn to it.* 

While in France Chasles alone interested himself 
thoroughly in its advancement, this new theory found 
its richest development in the third decade of the 
nineteenth century upon German soil, where almost 
at the same time the three great investigators, M6bius, 
Plucker, and Steiner entered the field. From this 
time on the synthetic and more constructive tendency 
followed by Steiner, von Staudt, and Mobius diverges*)" 

*Baltzer. t Brill, A., Antrittsrede in TUbtngen, 1884. 


from the analytic side of the modern geometry which 
Plucker, Hesse, Aronhold, and Clebsch had especially 

The Barycenirischer Calciil in the year 1827 fur- 
nished the first example of homogeneous co-ordinates, 
and along with them a symmetry in the developed 
formulae hitherto unknown to analytic geometry. In 
this calculus Mdbius started with the assumption that 
every point in the plane of a triangle ABC may be re- 
garded as the center of gravity of the triangle. In 
this case there belong to the points corresponding 
weights which are exactly the homogeneous co-ordi- 
nates of the point P with respect to the vertices of 
the fundamental triangle ABC. By means of this 
algorism M5bius found by algebraic methods a series 
of geometric theorems, for example those expressing 
invariant properties like the theorems on cross-ratios. 
These theorems, found analytically, Mobius sought to 
demonstrate geometrically also, and for this purpose 
he introduced with all its consequences the "law of 
signs " which expresses that for A^ B, C, points of a 
straight line, AB = ^BA, AB+ BA =0, AB+BC 
+ CA = 0, 

Independently of MObius, bnt starting from the same prin- 
ciples, Bellavitis came upon his new geometric method of equi- 
pollences.* Two equal and parallel lines drawn in the same direc- 
tion, AB and CD, are called equipollent (in Cayley's notation AB 
= CD). By this assumption the whole theory is reduced to the 

* Bellavitis, "Saggio di Applicazioni di un Nnovo Metodo di Geometria 
Analitica (Calcolo delle Equipollenze)," in Ann. Lamb. Ventto^ t. 5, 1835. 


consideration of segments proceeding from a ''xed point. Further 
it is assumed that AB + BC=AC (Addition). Finally for the seg- 
ments a, b, c, d, with inclinations a, p, y, 6 to a. fixed axis, the 

equation a=, -v must not only be a relation between lengths but 

must also show that a=sp-\-y — d (Proportion). For d=l and 
a = this becomes a^bc, i. e., the product of the absolute values 
of the lengths is a=bc and at the same time a^p-^-y (Multipli- 
cation). EquipoUence is therefore only a special case of the equal- 
ity of two objects, applied to segments.* 

MObius further introduced the consideration of 
correspondences of two geometric figures. The one- 
to-one correspondence, in which to every point of a 
first figure there corresponds one and only one point 
of a second figure and to every point of the second 
one and only one point of the first, Mobius called col- 
lineation. He constructed not only a coUinear image 
of the plane but also of ordinary space. 

These new and fundamental ideas which Mobius 
had laid down in the barycentric calculus remained 
for a long time almost unheeded and hence did not at 
once enter into the formation of geometric concep- 
tions. The works of Plucker and Steiner found a 
more favorable soil. The latter "had recognized in 
immediate geometric perception the sufficient means 
and the only object of his knowledge. Plucker, on 
the other hand,t sought his proofs in the identity of 
the analytic operation and the geometric construc- 

* Stolz, O., ViffrUsungen Uber allggmeine Artihmetik^ 1885-1886. 

t**Clebsch, Versuch einer Darlegung; und Wiirdigung seiner wissenschaft- 
lichen Leistungen von einigen seiner Freunde (Brill, Gordan, Klein, Liiroth, 
A. Mayer, NOther, Von der Muhll}," in Math. Ann., Bd. 7. 


tion, and regarded geometric truth only as one of the 
many conceivable antitypes of analytic relation/' 

At a later period (1855) Mobius engaged in the 
study of involutions of higher degree. Such an invo- 
lution of the mth degree consists of two groups each 
of m points : Ai, A2, Az, . . . A^ ; Bi, B^, B%y . . . B^^ 
which form two figures in such a way that to the 1st, 
2d, 3d, . . . m\\\ points of one group, as points of the 
first figure, there correspond in succession the 2d, 3d, 
4th . . . 1st points of the same group as points of the 
second figure, with the same determinate relation. In- 
volutions of higher degree had been previously studied 
by Poncelet (1843). He started from the theorem 
given by Sturm (1826), that by the conic sections of 
the surfaces of the second order « = 0, «^ = 0, u-^\v 
= 0, there are determined upon a straight line six 
points. A, A\ By B\ C, C* in involution, i. e., so that 
in the systems ABCA'BC and A' EC ABC not only 
A and A\ B and B\ C and C, but also A' and A, B 
and By C and C are corresponding point-pairs. This 
mutual correspondence of three point-pairs of a line 
Desargues had already (in 1639) designated by the 
term ** involution.*** 

Pliicker is the real founder of the modern analytic 
tendency, and he attained this distinction by "formu- 
lating analytically the principle of duality and follow- 
ing out its consequences. '*t His Analytisch-geometri- 
sche Uniersuchungen appeared in 1828. By this work 

♦ Baltzer. t Brill, A., Antrittsrede in TUbmgen, 1884. 


was created for geometry the method of symbolic no- 
tation and of undetermined coefiBcients, whereby one 
is freed from the necessity, in the consideration of the 
mutual relations of two figures, of referring to the 
system of co-ordinates, so that he can deal with the 
figures themselves. The System der analytischen Geo- 
meirie of 1835 furnishes, besides the abundant appli- 
cation of the abbreviated notation, a complete classi- 
fication of plane curves of the third order. In the 
Theorie der algebraischen Kurven of 1839, in addition 
to an investigation of plane curves of the fourth order 
there appeared those analytic relations between the 
ordinary singularities of plane curves which are gen- 
erally known as "Plucker's equations." 

These Plucker equations which at first are applied 
only to the four dualistically corresponding singulari- 
ties (point of inflexion, double point, inflexional tan- 
gent, double tangent) were extended by Cayley to 
curves with higher singularities. By the aid of devel- 
opments in series he derived four "equivalence num- 
bers " which enable us to determine how many singu- 
larities are. absorbed into a singular point of higher 
order, and how the expression for the deficiency of 
the curve is modified thereby. Cayley's results were 
confirmed, extended, and completed as to proofs by 
the works of NOther, Zeuthen, Halphen, and Smith. 
The fundamental question arising from the Cayley 
method of considering the subject, whether and by 
what change of parameters a curve with correspond- 


ing elementary singularities can be derived from a 
curve with higher singularity, for which the Pliicker 
and deficiency equations are the same, has been 
studied by A. Brill. 

Plucker's greatest service consisted in the intro- 
duction of the straight line as a space element. The 
principle of duality had led him to introduce, besides 
the point in the plane, the straight line, and in space 
the plane as a determining element. Pliicker also 
used in space the straight line for the systematic gen- 
eration of geometric figures. His first. works in this 
direction were laid before the Royal Society in Lon- 
don in 1865. They contained theorems on complexes, 
congruences, and ruled surfaces with some indications 
of the method of proof. The further development 
appeared in 1868 as Neue Geometrie des RaumeSy ge- 
griindet auf die Betrachtung der geraden Linie als Raum- 
element. Pliicker had himself made a study of linear 
complexes but his completion of the theory of com- 
plexes of the second degree was interrupted by death. 
Further extension of the theory of complexes was 
made by F. Klein. 

The results contained in Pliicker's last work have 
thrown a flood of light upon the difference between 
plane and solid geometry. The curved line of the 
plane appears as a simply infinite system either of 
points or of straight lines ; in space the curve may be 
regarded as a simply infinite system of points, straight 
lines or planes ; but from another point of view this 


curve in space may be replaced by the developable 
surface of which it is the edge of regression. Special 
cases of the curve in space and the developable sur- 
face are the plane curve and the cone. A further 
space figure, the general surface, is on the one side a 
doubly infinite system of points or planes, but on the 
other, as a special case of a complex, a triply infinite 
system of straight lines, the tangents to the surface. 
As a special case we have the skew surface or ruled 
surface. Besides this the congruence appears as a 
doubly,, the complex as a triply, infinite system of 
straight lines. The geometry of space involves a num- 
ber of theories to which plane geometry offers no anal- 
ogy. Here belong the relations of a space curve to 
the surfaces which may be passed through it, or of a 
surface to the gauche curves lying upon it. To the 
lines of curvature upon a surface there is nothing 
corresponding in the plane, and in contrast to the 
consideration of the straight line as the shortest line 
between two points of a plane, there stand in space 
two comprehensive and difficult theories, that of the 
geodetic line upon a given surface and that of the 
minimal surface with a given boundary. The ques- 
tion of the analytic representation of a gauche curve 
involves peculiar difficulties, since such a figure can 
be represented by two equations between the co-ordi- 
nates X, y, z only when the curve is the complete in- 
tersection of two surfaces. In just this direction tend 


the modem investigations of N6ther, Halphen, and 

Four years after the Analytisch-geometrische Unter- 
suchungen of Plucker, in the year 1832, Steiner pub- 
lished his Systemaiische Entwicklung der Abhdngigkeit 
geometrischer Gestalten, Steiner found the whole the- 
ory of conic sections concentrated in the single theo- 
rem (with its dualistic analogue) that a curve of the 
second order is produced as the intersection of two 
collinear or projective pencils, and hence the theory 
of curves and surfaces of the second order was essen- 
tially completed by him, so that attention could be 
turned to algebraic curves and surfaces of higher or- 
der. Steiner himself followed this course with good 
results. This is shown by the ** Steiner surface," and 
by a paper which appeared in 1848 in the Berliner 
Abhandlungen. In this the theory of the polar of a 
point with respect to a curved line was treated ex- 
haustively and thus a more geometric theory of plane 
curves developed, which was further extended by the 
labors of Grassmann, Chasles, Jonquiferes, and Cre- 

The names of Steiner and Pltlcker are also united in connec- 
tion with a problem which in its simplest form belongs to elemen- 
tary geometry, but in its generalization passes into higher fields. 
It is the Malfatti Problem, f In 1803 Malfatti gave out the following 
problem: From a right triangular prism to cut out three cylinders 
which shall have the same altitude as the prism, whose volumes 
shall be the greatest possible, and consequently the mass remain- 
♦Loria. \'^\XX%\.&itL^ Geschichte des Malfatti'' schen Probletns^ i^-ji. 


ing after their removal shall be a minimum. This problem he re- 
duced to what is now generally known as Malfatti's problem : In a 
given triangle to inscribe three circles so that each circle shall be 
tangent to two sides of the triangle and to the other two circles. He 
calculates the radii x^, X2, x^ of the circles sought in terms of the 
semi perimeter 5 of the triangle, the radius p of the inscribed cir- 
cle, the distances a^, a^, a^', b^, ^2, b^ of the vertices of the tri- 
angle from the center of the inscribed circle and its points of tan- 
gency to the sides, and gets : 

^2 = 3f-(^4-«2— P — ^8 — «i). 

^8 = 2^ (* + «8 — P — «1 — «2). 


without giving the calculation in full ; but he adds a simple con- 
struction. Steiner also studied this problem. He gave (without 
proof) a construction, showed that there are thirty-two solutions 
and generalized the problem, replacing the three straight lines by 
three circles. Pliicker also considered this same generalization. 
But besides this Steiner studied the same problem for space : In 
connection with three given conies upon a surface of the second 
order to determine three others which shall each touch two of the 
given conies and two of the required. This general problem re- 
ceived an analytic solution from Schellbach and Cayley, and also 
from Clebsch with the aid of the addition theorem of elliptic func- 
tions, while the more simple problem in the plane was attacked in 
the greatest variety of ways by Gergonne, Lehmus, Crelle, Grunert, 
Scheffler, Schellbach (who gave a specially elegant trigonometric 
solution) and Zorer. The first perfectly satisfactory proof of Stei- 
ner' s construction was given by Binder.* 

After Steiner came von Staudt and Chasles who 
rendered excellent service in the development of pro- 

* Programm Schifnthal, 1868. 


jective geometry. In 1837 Michel Chasles published 
his Aper^u historique sur Vorigine et le diveloppement 
des m^thodes en giom^trie^ a work in which both ancient 
and modern methods are employed in the derivation 
of many interesting results, of which several of the 
most important, among them the introduction of the 
cross-ratio (Chasles's "anharmonic ratio") and the 
reciprocal and collinear relation (Chasles's "duality" 
and '*homography"), are to be assigned in part to 
Steiner and in part to Mdbius 

Von Staudt's Geotnetrie der Lage appeared in 1847, 
his Beitrdge zur Geotnetrie der Lage^ 1856-1860 These 
works form a marked contrast to those of Steiner and 
Chasles who deal continually with metric relations 
and cross-ratios, while von Staudt seeks to solve the 
problem of "making the geometry of position an in- 
dependent science not standing in need of measure- 
ment." Starting from relations of position purely, 
von Staudt develops all theorems that do not deal 
immediately with the magnitude of geometric forms, 
completely solving, for example, the problem of the 
introduction of the imaginary into geometry. The 
earlier works of Poncelet, Chasles, and others had, 
to be sure, made use of complex elements but had 
defined the same in a manner more or less vague and, 
for example, had not separated conjugate complex 
elements from each other. Von Staudt determined 
the complex elements as double elements of involu- 
tion-relations. Each double element is characterized 


by the sense in which, by this relation, we pass from 
the one to the other. This suggestion of von Staudt's, 
however, did not become generally fruitful, and it 
was reserved for later works to make it more widely 
known by the extension of the originally narrow con- 

In the Beitrdge von Staudt has also shown how 
the cross ratios of any four elements of a prime form 
of the first class (von Staudt's Wiirfe) may be used to 
derive absolute numbers from pure geometry.* 

With the projective geometry is most closely con- 
nected the modern descriptive geometry. The former 
in its development drew its first strength from the 
considerations of perspective, the latter enriches itself 
later with the fruits matured by the cultivation of pro- 
jective geometry. 

The perspective of the Renaissance'f was devel- 
oped especially by French mathematicians, first by 
Desargues who used co-ordinates in his pictorial rep 
resentation of objects in such a way that two axes lay 
in the picture plane, while the third axis was normal 
to this plane. The results of Desargues were more 
important, however, for theory than for practice. 
More valuable results were secured by Taylor in his 
Linear Perspective (1715). In this a straight line is 
determined by its trace and vanishing point, a plane 
by its trace and vanishing line. This method was 

* Stolz, O., Vorlesungen Uber allgetneine Aritkmetik, 1885-1886. 
t Wiener. 


used by Lambert in an ingenious manner for different 
constructions, so that by the middle of the. eighteenth 
century even space-forms in general position could be 
pictured in perspective. 

Out of the perspective of the eighteenth century 
grew ''descriptive geometry/' first in a work of Fr6- 
zier's, which besides practical methods contained a 
special theoretical section furnishing proofs for all 
cases of the graphic methods considered. Even in 
the "description," or representation, Fr^zier replaces 
the central projection by the perpendicular parallel- 
projection, "which maybe illustrated by falling drops 
of ink."* The pipture of the plane of projection is 
called the ground plane or elevation according as the 
picture plane is horizontal or vertical. With the aid 
of this "description" Fr^zier represents planes, poly- 
hedra, surfaces of the second degree as well as inter- 
sections and developments. 

Since the time of Monge descriptive geometry has 
taken rank as a distinct science. The Lepons de g^o- 
mitrie descriptive (1795) form the foundation-pillars of 
descriptive geometry, since they introduce horizontal 
and vertical planes with the ground line and show 
how to represent points and straight lines by two pro- 
jections, and planes by two traces This is followed 
in the Le^ns by the great number of problems of in- 
tersection, contact and penetration which arise from 
combinations of planes with polyhedra and surfaces 

* wiener. 


of the second order. Mongers successors, Lacroix, 
Hachette, Olivier, and J. de la Gournerie applied 
these methods to surfaces of the second order, ruled 
surfaces, and the relations of curvature of curves and 

Just at this time, when the development of descriptive geom- 
etry in France had borne its first remarkable results, the technical 
high schools came into existence. In the year 1794 was established 
in Paris the Ecole CentrcUe des Travaux Publics from which in 
1795 the Acole Polytechnique was an outgrowth. Further techni- 
cal schools, which in course of time attained to university rank, 
were founded in Prague in 1806, in Vienna in 1815, in Berlin in 
1820, in Karlsruhe in 1825, in Munich in 1827, in Dresden in 1828, 
in Hanover in 1881, in Stuttgart 1882, in Ziirich in 1860, in 
Braunschweig in 1862, in Darmstadt in 1869, and in Aix-la-Chapelle 
in 1870. In these institutions the results of projective geometry 
were used to the greatest advantage in the advancement of descrip- 
tive geometry, and were set forth in the most logical manner by 
Fiedler, whose text-books and manuals, in part original and in 
part translations from the English, take a conspicuous place in the 
literature of the science. 

With the technical significance of descriptive geometry there 
has been closely related for some years an artistic side, and it is 
this especially which has marked an advance in works on axonom- 
etry (Weisbach, 1844), relief-perspective, photogrammetry, and 
theory of lighting. 

The second quarter of our century marks the time 
when developments in form-theory in connection with 
geometric constructions have led to the discovery of 
of new and important results. Stimulated on the one 
side by Jacobi, on the other by Poncelet and Steiner, 


Hesse (1837-1842) by an application of the transfor- 
mation of homogeneous forms treated the theory of 
surfaces of the second order and constructed their 
principal axes.* By him the notions of *' polar tri- 
angles" and "polar tetrahedra" and of "systems of 
conjugate points " were introduced as the geometric 
expression of analytic relations. To these were added 
the linear construction of the eighth intersection of 
three surfaces of the second degree, when seven of 
them are given, and also by the use of Steiner's theo- 
rems, the linear construction of a surface of the sec- 
ond degree from nine given points. Clebsch, follow- 
ing the English mathematicians, Sylvester, Cayley, 
and Salmon, went in his works essentially further than 
Hesse. His vast contributions to the theory of in- 
variants, his introduction of the notion of the defi- 
ciency of a curve, his applications of the theory of 
elliptic and Abelian functions to geometry and to the 
study of rational and elliptic curves, secure for him a 
pre-eminent place among those who have advanced 
the science of extension. As an algebraic instrument 
Clebsch, like Hesse, had a fondness for the theorem 
upon the multiplication of determinants in its appli- 
cation to bordered determinants. His worksf upon 
the general theory of algebraic curves and surfaces 

♦ NOther, "Otto Hesse," SchUhHtlch's ZeitaekHft, Bd. ao, HI. A. 

t " Clebsch, Versuch einer Darlegung and Wflrdignng seiner wlssen- 
schaftlichen Leistungen von einigen seiner Frennde" (Brill, Gordan, Klein, 
Liiroth, A. Mayer, NOther, Von der Miihll) Math. Ann., Bd. 7. 


began with the determination of those points upon an 
algebraic surface at which a straight h*ne has four- 
point contact, a problem also treated by Salmon but 
not so thoroughly. While now the theory of surfaces 
of the third order with their systems of twenty-seven 
straight lines was making headway on English soil, 
Clebsch undertook to render the notion of "defi- 
ciency*' fruitful for geometry. This notion, whose 
analytic properties were not unknown to Abel, is found 
first in Riemann's Theorie der AbeVschen Funktionen 
(1857). Clebsch speaks also of the deficiency of an 
algebraic curve of the «th order with d double points 
and r points of inflexion, and determines the number 
/ = J(« — 1)(« — 2) — d — r. To one class of plane 
or gauche curves characterized by a definite value of 
p belong all those that can be made to pass over into 
one another by a rational transformation or which 
possess the property that any two have a one-to-one 
correspondence. Hence follows the theorem that only 
those curves that possess the same 3/— 8 parameters 
(for curves of the third order, the same one parame- 
ter) can be rationally transformed into one another. 

The diflficult theory of gauche curves* owes its first 
general results to Cayley, who obtained formulae cor- 
responding to Plucker's equations for plane curves. 
Works on gauche curves of the third and fourth orders 
had already been published by M5bius, Chasles, and 
Von Staudt. General observations on gauche curves 

* Loria. 


in more recent times are found in theorems of NOther 
and Halphen. 

The foundations of enumerative geometry* are 
found in Chasles's method of characteristics (1864). 
Chasles determined for rational configurations of one 
dimension a correspondence-formula which in the 
simplest case may be stated as follows : If two ranges 
of points ^1 and ^s lie upon a straight line so that to 
every point x of ^1 there correspond in general a 
points J' in ^j, and again to every point j' of J?2 there 
always correspond /8 points x in ^1, the configuration 
formed from R\ and R^ has (a -f- /8) coincidences or 
there are (a -f- /8) times in which a point x coincides 
with a corresponding point y. The Chasles corre- 
spondence-principle was extended inductively by Cay- 
ley in 1866 to point-systems of a curve of higher 
deficiency and this extension was proved by BrilKf 
Important extensions of these enumerative formulae 
(correspondence-formulae), relating to general alge- 
braic curves, have been given by Brill, Zeuthen, and 
Hurwitz, and set forth in elegant form by the intro- 
duction of the notion of deficiency. An extended 
treatment of the fundamental problem of enumerative 
geometry, to determine how many geometric config- 
urations of given definition satisfy a sufficient number 
of conditions, is contained in the Kalkiil der abzdhlen- 
den Geomeirie by H. Schubert (1879). 

The simplest cases of one-to-one correspondence 

*Loria. \ Matkem. AnnalentVX, 


or uniform representation, are furnished by two planes 
s'iperimposed one upon the other. These are the 
s milarity studied by Poncelet and the collineation 
treated by Mobius, Magnus, and Chasles.* In both 
cases to a point corresponds a point, to a straight line 
a straight line. From these linear transformations 
Poncelet, Plucker, Magnus, Steiner passed to the 
quadratic where they first investigated one-to-one cor- 
respondences between two separate planes. The 
'* Steiner projection" (1832) employed two planes £j 
and £2 together with two straight lines gi and g^ not 
CO- planar. If we draw through a point Pi or /j of £i 
or £2 the straight line xi or xt which cuts gi as well 
as g2i and determines the intersection Xf or Xi, with 
£f or ^1, then are Pi and X2, and P^ and Xi corre- 
sponding points. In this manner to every straight 
line of the one plane corresponds a conic section in 
the other. In 1847 Plucker had determined a point 
upon the hyperboloid of one sheet, like fixing a point 
in the plane, by the segments cut off upon the two 
generators passing through the point by two fixed 
generators. This was an example of a uniform rep- 
resentation of a surface of the second order upon the 

The one-to-one relation of an arbitrary surface of 
the second order to the plane was investigated by 
Chasles in 1863, and this work marks the beginning 
of the proper theory of surface representation which 

* Loria. 


found its further development when Clebsch and Cre 
mona independently succeeded in the representation 
of surfaces of the third order. Cremona's important 
results were extended by Cayley, Clebsch, Rosanes, 
and NGther, to the last of whom we owe the impor- 
tant theorem that every Cremona transformation which 
as such is uniform forward and backward can be 
effected by the repetition of a number of quadratic 
transformations. In the plane only is the aggregate 
of all rational or Cremona transformations known ; 
for the space of three dimensions, merely a beginning 
of the development of this theory has been made.* 

A specially important case of one-to-one corre- 
spondence is that of a conformal representation of a 
surface upon the plane, because here similarity in the 
smallest parts exists between original and image. The 
simplest case, the stereographic projection, was known 
to Hipparchus and Ptolemy. The representation by 
reciprocal radii characterized by the fact that any two 
corresponding points P\ and P\ lie upon a ray through 
the fixed point O so that 0P\' ^-Pj = constant, is also 
conformal. Here every sphere in space is in general 
transformed into a sphere. This transformation, stud- 
ied by Bellavitis 1836 and Stubbs 1843, is especially 
useful in dealing with questions of mathematical phys- 
ics. Sir Wm. Thomson calls it **the principle of elec- 
tric images." The investigations upon representa- 

* Klein, F., Vergleichende Betrachtungen Uder ncucre geometrische Forsch- 
u ft gen, 1872. 


tions, made by Lambert and Lagrange, but more 
especially those by Gauss, lead to the theory of curva- 

A further branch of geometry, differential geom- 
etry (as applied curvature of surfaces), considers in 
general not first the surface in its totality but the 
properties of the same in the neighborhood of an or- 
dinary point of the surface, and with the aid of the 
differential calculus seeks to characterize it by ana- 
lytic formulae. 

The first attempts to enter this domain were made 
by Lagrange (1761), Euler (1766), and Meusnier(1776). 
The former determined the differential equation of 
minimal surfaces ; the two latter discovered certain 
theorems upon radii of curvature and surfaces of cen- 
ters. But of fundamental importance for this rich do- 
main have been the investigations of Monge, Dupin, 
and especially of Gauss. In the Application de Vana- 
lyse d la g^omitrie (1795), Monge discusses families 
of surfaces (cylindrical surfaces, conical surfaces, and 
surfaces of revolution, — envelopes 'with the new no- 
tions of characteristic and edge of regression) and de- 
termines the partial differential equations distinguish- 
ing each. In the year 1813 appeared the Diveloppements 
de giomitrie by Dupin. It introduced the indicatrix 
at a point of a surface, as well as extensions of the 
theory of lines of curvature (introduced by Monge) 
and of asymptotic curves. 

Gauss devoted to differential geometry three trea- 


. the most celebrated, Disquisiiiones gemraUs circa 
/erficics curvas, appeared in 1827, the other two 
(Inter sue hungen iiber Gegensidnde der hoheren Geoddsie 
were published in 1843 and 1846. In the Disquisi- 
iiones ^ to the preparation of which he was led by his 
own astronomical and geodetic investigations,* the 
spherical representation of a surface is introduced. 
The one-to-one correspondence between the surface 
and the sphere is established by regarding as corre- 
sponding points the feet of parallel normals, where 
obviously we must restrict ourselves to a portion of 
the given surface, if the correspondence is to be main- 
tained. Thence follows the introduction of the curvi- 
linear co-ordinates of a surface, and the definition of 
the measure of curvature as the reciprocal of the pro- 
duct of the two radii of principal curvature at the 
point under consideration. The measure of curvature 
is first determined in ordinary rectangular co-ordinates 
and afterwards also in curvilinear co-ordinates of the 
surface. Of the latter expression it is shown that it is 
not changed by any bending of the surface without 
stretching or folding (that it is an invariant of curva- 
ture). Here belong the consideration of geodetic 
lines, the definition and a fundamental theorem upon 
the total curvature {curvatura integrd) of a triangle 
bounded by geodetic lines. 

The broad views set forth in the Disquisiiiones of 
1827 sent out fruitful suggestions in the most vari- 

« Brill. A.. Antritttrtd* in Tubingen, 1884. 


ous directions. Jacobi determined the geodetic lines 
of the general ellipsoid. With the aid of elliptic co- 
ordinates (the parameters of three surfaces of a sys- 
tem of confocal surfaces of the second order passing 
through the point to be determined) he succeeded in 
integrating the partial differential equation so that the 
equation of the geodetic line appeared as a relation 
between two Abelian integrals. The properties of the 
geodetic lines of the ellipsoid are derived with espe- 
cial ease from the elegant formulae given by Liou- 
ville. By Lam6 the theory of curvilinear co-ordinates, 
of which he had investigated a special case in 1837, 
was developed in 1859 into a theory for space in his 
LtQons sur la thiorie des coordonnies curvilignes. 

The expression for the Gaussian measure of curva- 
ture as a function of curvilinear co-ordinates has given 
an impetus to the study of the so called differential 
invariants or differential parameters. These are cer- 
tain functions of the partial derivatives of the coeflB- 
cients in the expression for the square of the line- ele- 
ment which in the transformation of variables behave 
like the invariants of modern algebra. Here Sauc6, 
Jacobi, C. Neumann, and Halphen laid the founda- 
tions, and a general theory has been developed b> 
Beltrami.* This theory, as well as the contact- trans- 
formations of Lie, moves along the border line be- 
tween geometry and the theory of differential equa- 
tions, f 

♦ Mem. di Bologna, VIII. t Loria. 


With problems of the mathematical theory of light are con- 
nected certain investigations upon systems of rays and the prop- 
erties of infinitely thin bundles of rays, as first carried on by Du- 
pin, Mains. Ch. Sturm, Bertrand, Transon, and Hamilton. The 
celebrated works of Kummer (1857 and 1866) perfect Hamilton's 
results upon bundles of rays and consider the number of singular- 
ities of a system of rays and its focal surface. An interesting ap 
plication to the investigation of the bundles of rays between the 
lens and the retina, founded on the study of the infinitely thin 
bundles of normals of the ellipsoid, was given by O. BOklen.* 

Non- Euclidean Geometry. — Though the respect 
which century after century had paid to the Elements 
of Euclid was unbounded, yet mathematical acuteness 
had discovered a vulnerable point ; and this point f 
forms the eleventh axiom (according to Hankel, reck- 
oned by Euclid himself among the postulates) which 
affirms that two straight lines intersect on that side of 
a transversal on which the sum of the interior angles 
is less than two right angles. Toward the end of the 
last century Legendre had tried to do away with this 
axiom by making its proof depend upon the others, but 
his conclusions were invalid. This effort of Legendre's 
was an indication of the search now beginning after a 
geometry free from contradictions, a hyper- Euclidean 
geometry or pangeometry. Here also Gauss was 
among the first who recognized that this axiom could 
not be proved. Although from his correspondence 
with Wolfgang Bolyai and Schumacher it can easily 

* Kronecker's Journal, Band 96. Forischritte, 1884. 
t Loria. 


be seen that he had obtained some definite results in 
this field at an early period, he was unable to decide 
upon any further publication. The real pioneers in 
the Non-Euclidean geometry were Lobachevski and 
the two Bolyais. Reports of the investigations of 
Lobachevski first appeared in the Courier of Kasan, 
1829-1830, then in the transactions of the Univer- 
sity of Kasan, 1835-1839, and finally as Geometrische 
Untersuchungen fiber die Theorie der Parallellinien^ 1840, 
in Berlin. By Wolfgang Bolyai was published (1832- 
1833 *) a two- volume work, Tent amen Juventutem siu- 
diosatn in eletnenta Matheseos purae, etc. Both works 
were for the mathematical world a long time as good 
as non-existent till first Riemann, and then (in 1866) 
R. Baltzer in his Elemente, referred to Bolyai. Almost 
at the same time there followed a sudden mighty ad- 
vance toward the exploration of this ''new world" by 
Riemann, Helmholtz, and Beltrami. It was recog- 
nized that of the twelve Euclidean axioms f nine are 
of essentially arithmetic character and therefore hold 
for every kind of geometry ; also to every geometry is 
applicable the tenth axiom upon the equality of all 
right angles. The twelfth axiom (two straight lines, 
or more generally two geodetic lines, include no 
space) does not hold for geometry on the sphere. 
The eleventh axiom (two straight lines, geodetic 

* Schmidt, "Ausdem Leben zweier nngarischen Mathematiker," Grunert 
Arch,, Bd. 48. 

t Brill, A., Ugber da* t^t Axiom des Euclid, 1883. 


lines, intersect when the sum of the interior angles is 
less than two right angles) does not hold for geometry 
on a pseudo-sphere, but only for that in the plane. 

Riemann, in his paper **Ueber die Hypothesen, 
welche der Geometric zu Grunde liegen,"* seeks to 
penetrate the subject by forming the notion of a mul- 
tiply extended manifoldness ; and according to these 
investigations the essential characteristics of an «-ply 
extended manifoldness of constant measure of curva- 
ture are the following : 

1. ** Every point in it may be determined by n 
variable magnitudes (co-ordinates). 

2. <*The length of a line is independent of posi- 
tion and direction, so that every line is measurable 
by every other. 

3. *'To investigate the measure-relations in such 
a manifoldness, we must for every point represent the 
line-elements proceeding from it by the corresponding 
differentials of the co-ordinates. This is done by virtue 
of the hypothesis that the length-element of the line 
is equal to the square root of a homogeneous function 
of the second degree of the differentials of the co- 
ordinates. " 

At the same time Helmholtzf published in the 
**Thatsachen, welche der Geometric zu Grunde lie- 
gen," the following postulates : 

« G9ttinger Ahhandlungen, XIII, x868. Fortschritte, x868. 
\FortschrHte, i868. . 


1. **A point of an ;i-tuple manifoldness is deter- 
mined by n CO ordinates. 

2. "Between the 2n co-ordinates of a point-pair 
there exists an equation, independent of the move- 
ment of the latter, which is the same for all congruent 

3. ''Perfect mobility of rigid bodies is assumed. 

4. ** If a rigid body of n dimensions revolves about 
n — 1 fixed points, then revolution without ireversal 
will bring it back to its original position." 

Here spatial geometry has satisfactory foundations 
for a development free from contradictions, if it is fur- 
ther assumed that space has three dimensions and is 
of unlimited extent. 

One of the most surprising results of modem geo- 
metric investigations was the proof of the applicabil- 
ity of the non-Euclidean geometry to pseudo-spheres 
or surfaces of constant negative curvature.* On a 
pseudo-sphere, for example, it is true that a geodetic 
line (corresponding to the straight line in the plane, 
the great circle on the sphere) has two separate points 
at infinity; that through a point P, to a given geodetic 
line gt there are two parallel geodetic lines, of which, 
however, only one branch beginning at P cuts g at in- 
finity while the other branch does not meet g at all ; 
that the sum of the angles of a geodetic triangle is 
less than two right angles. Thus we have a geometry 
upon the pseudo- sphere which with" the spherical ge- 

♦ Cayley, Address to the British Association, etc,, 1883. 


ometry has a common limiting case in the ordinary 
or Euclidean geometry. These three geometries have 
this in common that they hold for surfaces of constant 
curvature. According as the constant value of the 
':urvature is positive, zero, or negative, we have to do 
with spherical, Euclidean, or pseudo-spherical geom- 

A new presentation of the same theory is due to 
F. Klein. After projective geometry had shown that 
in projection or linear transformation all descriptive 
properties and also some metric relations of the fig- 
ures remain unaltered, the endeavor was made to find 
for the metric properties an expression which should 
remain invariant after a linear transformation. After 
a preparatory work of Laguerre which made the *'no 
tion of the angle projective," Cayley, in 1859, found the 
general solution of this problem by considering ''every 
metric property of a plane figure as contained in a 
projective relation between it and a fixed conic." 
Starting from the Cayley theory, on the basis of the 
consideration of measurements in space, Klein suc- 
ceeded in showing that from the projective geometry 
with special determination of measurements in the 
plane there could be derived an elliptic, parabolic, 
or hyperbolic geometry,* the same fundamentally as 
the spherical, Euclidean, or pseudo- spherical geom- 
etry respectively. 

The need of the greatest possible generalization 

* Forischritte, 1871. 


and the continued perfection of the analytic apparatus 
have led to the attempt to build up a geometry of n 
dimensions; in this, however, only individual relations 
have been considered. Lagrange* observes that ''me- 
chanics may be regarded as a geometry of four dimen- 
sions.'* Pliicker endeavored to clothe the notion of 
arbitrarily extended space in a form easily understood. 
He showed that for the point, the straight line or the 
sphere, the surface of the second order, as a space 
element, the space chosen must have three, four, or 
nine dimensions respectively. The first investigation, 
giving a different conception from Plucker's and ''con- 
sidering the element of the arbitrarily extended mani- 
foldness as an analogue of the point of space," is 
foundf in H. Grassmann's principal work, Die Wissen- 
schaft der exiensiven Gross e oder die lineale Ausdehnungs- 
lehre (1844), which remained almost wholly unno- 
ticed, as did his Geometrische Analyse (1847). Then 
followed Riemann*s studies in multiply extended mani- 
foldnesses in his paper Ueber die Hypoihesen, etc., and 
they again furnished the starting point for a series of 
modern works by Veronese, H. Schubert, F. Meyer, 
Segre, Castelnuovo, etc. 

A Geomeiria situs in the broader sense was created 
by Gauss, at least in name; but of it we know scarcely 
more than certain experimental truths. % The Analysis 

* Loria. 

t F. Klein, Vergleichende Bctrachtuugeu Hber neuere geometrische For- 
s hungen, 1872. 

t Brill, A., Antrittsrede in Tubingen, 1884. 


situsy suggested by Riemann, seeks what remains fixed 
after transformations consisting of the combination of 
infinitesimal distortions.* This aids in the solution 
of problems in the theory of functions. The contact- 
transformations already considered by Jacobi have 
been developed by Lie. A contact- transformation is 
defined analytically by every substitution which ex- 
presses the values of the co-ordinates Xy y, z, and the 

partial derivatives -z- =/, -j- =^, in terms of quan- 
tities of the same kind, y, y, »*, /', /. In such a 
transformation contacts of two figures are replaced by 
similar contacts. 

Also a ** geometric theory of probability" has been 
created by Sylvester and Woolhouse;f Crofton uses 
it for the theory of lines drawn at random in space. 

In a history of elementary mathematics there pos- 
sibly calls'for attention a related field, which certainly 
cannot be regarded as a branch of science, but yet 
which to a certain extent reflects the development of 
geometric science, the history of geometric illustrative 
material J Good diagrams or models of systems of 
space- elements assist in teaching and have frequently 
led to the rapid spread of new ideas. In fact in the 
geometric works of Euler, Newton, and Cramer are 
found numerous plates of figures. Interest in the 

♦ F.Klein. ^ FortscArttte, 1B6S. 

t Brill, A., Ueber die Mode llsamtti lung des ntathematischen Seminars der 
Universitat TUbittgen, 1886. Mathematisch-naturwissenschaftlich* Mitthei- 
lungen von O. BSklen. 1887. 


construction of models seems to have been manifested 
first in France in consequence of the example and ac- 
tivity of Monge. In the year 1830 the Conservatoire 
des arts et mitiers in Paris possessed a whole series of 
thread models of surfaces of the second degree, con. 
oids and screw surfaces. A further advance was made 
by Bardin (1855). He had plaster and thread mod- 
els constructed for the explanation of stone-cutting, 
toothed gears and other matters. His collection was 
considerably enlarged by Muret. These works of 
French technologists met with little acceptance from 
the mathematicians of that country, but, on the con- 
trary, in England Cayley and Henrici put on exhibi- 
tion in London in 1876 independently constructed 
models together with other scientific apparatus of the 
universities of London and Cambridge. 

In Germany the construction of models experi- 
enced an advance from the time when the methods of 
projective geometry were introduced into descriptive 
geometry. Plucker, who in his drawings of curves of 


the third order had in 1835 showed his interest in re- 
lations of form, brought together in 1868 the first 
large collection of models. This consisted of models 
of complex surfaces of the fourth order and was con- 
siderably enlarged by Klein in the same field. A 
special surface of the fourth order, the wave-surface 
for optical bi-axial crystals was constructed in 1840 
by Magnus in Berlin, and by Soleil in Paris. . In the 
year 1868 appeared the first model of a surface of the 


third order with its twenty-seven straight lines, by 
Chr. Wiener. In the sixties, Kummer constructed 
models of surfaces of the fourth order and of certain 
focal surfaces. His pupil SchWarz likewise constructed 
a series of models, among them minimal surfaces and 
the surfaces of centers of the ellipsoid. At a meeting 
of mathematicians in Gdttingen there was made a 
notable exhibition of models which stimulated further 
work in this direction. 

In wider circles the works suggested by A. Brill, 
F. Klein, and W. Dyck in the mathematical seminar 
of the Munich polytechnic school have found recogni- 
tion. There appeared from 1877 to 1890 over a hun- 
dred models of the most various kinds, of value not 
only in mathematical teaching but also in lectures en 
perspective, mechanics and mathematical physics. 

In other directions also has illustrative material of 
this sort been multiplied, such as surfaces of the third 
order by Rodenberg, thread models of surfaces and 
gauche curves of the fourth order by Rohn, H.Wiener, 
and others. 

♦ ♦ 

If one considers geometric science as a whole, it 
cannot be denied that in its field no essential differ- 
ence between modern analytic and modern synthetic 
geometry any longer exists. The subject-matter and 
the methods of proof in both directions have gradu- 
ally taken almost the same form. Not only does the 
synthetic method make use of space intuition; the 


analytic representations also are nothing less than a 
clear expression of space relations. And since metric 
properties of figures may be regarded as relations of 
the same to a fundamental form of the second order, 
to the great circle at infinity, and thus can be brought 
into the aggregate of projective properties, instead of 
analytic and synthetic geometry, we have only a pro- 
jective geometry which takes the first place in the 
science of space.* 

The last decades, especially of the development of 
German mathematics, have secured for the science a 
leading place. In general two groups of allied works 
may be recognized. f In the treatises of the one ten- 
dency »* after the fashion of a Gauss or a Dirichlet, 
the inquiry is concentrated upon the exactest possible 
limitation of the fundamental notions " in the theory 
of functions, theory of numbers, and mathematical 
physics. The investigations of the other tendency, 
as is to be seen in Jacobi and Clebsch, start ''from a 
small circle of already recognized fundamental con- 
cepts and aim at the relations and consequences which 
spring from them," so as to serve modem algebra and 

On the whole, then, we may say that]; "mathe- 
matics have steadily advanced from the time of the 
Greek geometers. The achievements of Euclid, Archi- 
medes, and ApoUonius are as admirable now as they 

* F.Klein. t Clebsch. 

t Cay ley, A., Acldreu to the British Asson'ation , etc., 1883. 


were in their own days. Descartes's method 'of co- 
ordinates is a possession forever. But mathematics 
have never been cultivated more zealously and dili- 
gently, or with greater success, than in this century — 
in the last half of it, or at the present time : the ad- 
vances made have been enormous, the actual field is 
boundless, the future is full of hope." 




T^RIGONOMETRY was developed by the ancients 
^ for purposes of astronomy. In the first period a 
number of fundamental formulae of trigonometry were 
established, though not in modern form, by the Greeks 
and Arabs, and employed in calculations. The second 
period, which extends from the time of the gradual 
rise of mathematical sciences in the earliest Middle 
Ages to the middle of the seventeenth century, estab- 
lishes the science of calculation with angular func- 
tions and produces tables in which the sexagesimal 
division is replaced by decimal fractions, which marks 
a great advance for the purely numerical calculation. 
During the third period, plane and spherical trigo- 
nometry develop, especially polygonometry and poly- 
hedrometry which are almost wholly new additions to 
the general whole. Further additions are the projec- 
tive formulae which have furnished a series of inter- 
esting results in the closest relation to projective ge- 





The Papyrus of Ahmes* speaks of a quotient 
called seqt. After observing that the great pyramids 
all possess approximately equal angles of inclination, 
the assumption is rendered probable that this seqt is 
identical with the cosine of the angle which the edge 
of the pyramid forms with the diagonal of the square 
base. This angle is usually 52^. In the Egyptian 
monuments which have steeper sides, the seqt ap- 
pears to be equal to the trigonometric tangent of the 
angle of inclination of one of the faces to the base. 

Trigonometric investigations proper appear first 
among the Greeks. Hypsicles gives the division of 
,the (nrcumference into three hundred sixty degrees, 
which, indeed, is of Babylonian origin but was first 
turned to advantage by the Gr-ceks. After the intro- 
duction of this division of the circle, sexagesimal 
fractions were to be found in all the astronomical cal- 
culations of antiquity (with the single exception of 
Heron), till finally Peurbach and Regiomontanus pre- 
pared the way for the decimal reckoning. Hipparchus 
was the first to complete a table of chords, but of this 
we have left only the knowledge of its former exist- 

♦Cantor, I., p. 58. 


ence. In Heron are found actual trigonometric for- 
mulae with numerical ratios for the calculation of the 
areas of regular polygons and in fact all the values of 

cotf — jfor « = 3, 4, ... 11, 12 are actually computed.* 

Menelaus wrote six books on the calculation of chords, 
but these, like the tables of Hipparchus, are lost. On 
the contrary, three books of the Spherics of Menelaus 
are known in Arabic and Hebrew translations. These 
contain theorenis on transversals and on the congru- 
ence of spherical as well as plane triangles, and for 
the spherical triangle the theorem that a'\-b-\'C<i 4JP, 
t» + )8+r>2J?. 

The most important work of Ptolemy consists in 
the introduction of a formal spherical trigonometry 
for astronomical purposes. The thirteen books of the 
Great Collection which contain the Ptolemaic astron- 
omy and trigonometry were translated into Arabic, 
then into Latin, and in the latter by a blending «f the 
Arabic article al with a Greek word arose the won 
Almagest^ now generally applied to the great work of 
Ptolemy. Like Hypsicles, Ptolemy also, after the 
ancient Babylonian fashion, divides the circumfer- 
ence into three hundred sixty degrees, but he, in ad 
dition to this, bisects every degree. As something 
new we find in Ptolemy the division of the diameter 
of the circle into one hundred twenty equal parts, 
from which were formed after the sexagesimal fashion 

♦Tannery in Mint. Bord., 1881. 


two classes of subdivisions. ' In the later Latin trans- 
lations these sixtieths of the first and second kind 
were called respectively partes tninuiae primae and 
partes minutae secundae. Hence came the later terms 
"minutes" and ** seconds." Starting from his theo- 
rem upon the inscribed quadrilateral, Ptolemy calcu 
lates the chords of arcs at intervals of half a degree. 
But he develops also some theorems of plane and 
especially of spherical trigonometry, as for example 
theorems regarding the right angled spherical tri- 

A further not unimportant advance in trigonom- 
etry is to be noted in the works of the Hindus. The 
division of the circumference is the same as that of 
the Babylonians and Greeks ; but beyond that there 
is an essential deviation. The radius is not divided 
sexagesimally after the Greek fashion, but the arc of 
the same length as the radius is expressed in min- 
utes ; thus for the Hindus r = 3438 minutes. Instead 
of the whole chords {jiva), the half chords {ardhajya 
are put into relation with the arc. In this relation of 
the half- chord to the arc, though known to the Greeks, 
we recognize an important advance among the Hindus. 
In accordance with this notion they were therefore 
familiar with what we now call the sine of an angle. 
Besides this they calculated the ratios corresponding 
to the versed sine and the cosine and gave them spe- 
cial names, calling the versed sine uikramajya^ the 
cosine koiij'ya. They also knew the formula sin*^ 


-|- cos^a = 1 . They did not, however, apply their 
trigonometric knowledge to the solution of plane tri- 
angles, but with them trigonometry was inseparably 
connected with astronomical calculations. 

As in the rest of mathematical science, so in trig- 
onometry, were the Arabs*pupils of the Hindus, and 
still more of the Greeks, but not without important 
devices of their own. To Al Battani it was well known 
that the. introduction of half chords instead of whole 
chords, as these latter appear in the Almagest, and 
therefore reckoning with the sine of an angle, is of 
essential advantage in the applications. In addition 
to the formulae found in the Almagest^ Al Battani 
gives the relation, true for the spherical triangle, 
cos^j = cos^cos^ + sin^sin^cosa. In the considera- 
tion of right-angled triangles in connection with 

shadow-measuring, we find the quotients and 

cos a 

— : — . These were reckoned for each degreee by Al 
sin a 

Battani and arranged in a small table. Here we find 
the beginnings of calculation with tangents and co- 
tangents. These names, however, were introduced 
much later. The origin of the term "sine" is due to 
Al Battani. His work upon the motion of the stars 
was translated into Latin by Plato of Tivoli, and this 
translation contains the word j/V^^^j for half chord.* 
In Hindu the half chord was called ardhajya or also 
jiva (which was used originally only for the whole 

* Cantor, I , p. 693. This account is now known to be incorrect. 


chord); the latter word the Arabs adopted, simply 
by reason of its sound, as jiba. The consonants of 
this word, which in Arabic has no meaning of its 
own, might be read /V7/3 = bosom, or incision, and 
this pronunciation, which apparently was naturalized 
comparatively soon by the Arabs, Plato of Tivoli 
translated properly enough into sinus.* Thus was in- 
troduced the first of the modern names of the trigo- 
nometric functions. 

Of astronomical tables there was no lack at that 

time. Abul Wafa, by whom the ratio was called 

cos a 

the "shadow" belonging to the angle a, calculated a 
table of sines at intervals of half a degree and also a 
table of tangents, which however was used only for 
determining the altitude of the sun. About the same 
time appeared the hakimitic table of sines which Ibn 
Yunus of Cairo was required to construct by direction 
of the Egyptian ruler Al Hakim. 

Among the Western Arabs the celebrated astron- 
omer Jabir ibn Aflah, or Geber, wrote a complete trigo- 
nometry (principally spherical) after a method of his 
own, and this work, rigorous throughout in its proofs, 
was published in the Latin edition of his Astronomy 
by Gerhard of Cremona. This work contains a col- 
lection of formulae upon the right-angled spherical 
triangle. In the plane trigonometry he does not go 

♦Gerhard of Cremona used the word sinus. 


beyond the Almagest, and hence he here deals only 
with whole chords, just as Ptolemy had taught. 



Of the mathematicians outside of Germany in this 
period, Vieta made a most important advance by his 
introduction of the reciprocal triangle of a spherical 
triangle. In Germany the science was advanced by 
Regiomontanus and in its elements was presented with 
such skill and thorough knowledge that the plan laid 
out by him has remained in great part up to the pres- 
ent day. Peurbach had already formed the plan of 
writing a trigonometry but was prevented by death. 
Regiomontanus was able to carry out Feurbach's idea 
by writing a complete plane and spherical trigonom- 
etry. After a brief geometric introduction Regiomon- 
tanus's trigonometry begins with the right-angled tri- 
angle, the formulae needed for its computation being 
derived in terms of the sine alone and illustrated by 
numerical examples. The theorems on the right- 
angled triangle are used for the calculation of the 
equilateral and isosceles triangles. Then follow the 
principal cases of the oblique angled triangle of which 
the first (a from «, 3, c) is treated with much detail. 
The second book contains the sine theorem and a 



series of problems relating to triangles. The third, 
fourth, and fifth books bring in spherical trigonometry 
with many resemblances to Menelaus ; in particular 
the angles are found from the sides. The case of the 
plane triangle (a from «, b, c), treated with consider- 
able prolixity by Regiomontanus, received a shorter 
treatment from Rhaeticus, who established the for- 
mula coti^a= , where p is the radius of the in- 
scribed circle. 

In this period were also published Napier's equa- 
tions, or analogies. They express a relation between 
the sum or difference of two sides (angles) and the 
third side (angle) and the sum or difference of the two 
opposite angles (sides). 

Of modern terms, as already stated, the word 
**sine" is the oldest. About the end of the sixteenth 
century, or the beginning of the seventeenth, the ab- 
breviation cosine for complemenii sinus was intrc duced 
by the Englishman Gunter (died 1626). The terms 
tangent and secant were first used by Thomas Finck 
(1583); the term versed sine was used still earlier.* 

By some writers of the sixteenth century, e. g., by 
Apian, sinus rectus secundus was written instead of co- 
sine. Rhaeticus and Vieta have perpendiculum and 
basis for sine and cosine.f The natural values of the 
cosine, whose logarithms were called by Kepler **anti- 

♦Baltzer, R., Die Elemente der Mathematik, 1885. 
tPfleiderer, Trigonometries 1802. 


logarithms," are first found calculated in the trigo- 
nometry of Copernicus as published by Rhaeticus.* 

The increasing skill in practical computation, and 
the need of more accurate values for astronomical 
purposes, led in the sixteenth century to a strife after 
the most complete trigonometric tables possible. The 
preparation of these tables, inasmuch as the calcula- 
tions were made without logarithms, was very tedious. 
Rhaeticus alone had to employ for this purpose a 
number of computers for twelve years and spent 
thereby thousands of gulden, f 

The first table of sines of German origin is due to 
Peurbach. He put the radius equal to 600 000 and 
computed at intervals of 10' (in Ptolemy r = 60, with 
some of the Arabs r = 150). Regiomontanus com- 
puted two new tables of sines, one for r=6 000 000, 
the other, of which no remains are left, for r = 
10 000 000. Besides these we have from Regiomon- 
tanus a table of tangents for every degree, r= 100000. 
The last two tables evidently show a transition from 
the sexagesimal system to the decimal. A table of 
sines for every minute, with r= 100 000, was pre- 
pared by Apian. 


In this field should also be mentioned the indefat- 
igable perseverance of Joachim Rhaeticus. He did 
not associate the trigonometric functions with the 
arcs of circles, but started with the right-angled tri- 

*M. Curtze, in SchlSmilch' s Zeitschrtft, Bd. XX. 

1 Gerhardt, Geschickte der Mathematik in Deuisckland, 1877. 


angle and used the terms perpendiculum for sine, basis 
for cosine. He calculated (partly himself and partly 
by the help of others) the first table of secants ; later, 
tables of sines, tangents, and secants for every 10", 
for radius =10 000 millions, and later still, for r = 
10". After his death the whole work was published 
by Valentin Otho in the year 1596 in a volume of 1468 

To the calculation of natural trigonometric func- 
tions Bartholomaeus Pitiscus also devoted himself. 
In the second book of his Trigonometry he sets forth 
his views on computations of this kind. His tables 
contain values of the sines, tangents, and secants on 
the left, and of the complements of the sines, tangents 
and secants (for so he designated the cosines, cotan- 
gents, and cosecants) on the right. There, are added 
proportional parts for 1', and even for 10". In the 
whole calculation r is assumed equal to lO**. The 
work of Pitiscus appeared at the beginning of the 
seventeenth century. 

The tables of the numerical values of the trigono- 
metric functions had now attained a high degree of 
accuracy, but their real significance and usefulness 
were first shown by the introduction of logarithms. 

Napier is usually regarded as the inventor of log- 
arithms, although Cantor's review of the evidencef 
leaves no room for doubt that Biirgi was an indepen- 
dent discoverer. His Progress Tabulen^ computed be- 

* Gerhardt. t Cantor, II., pp. 662 et seq. 


tween 1603 and 1611 but not published until 1620 is 
really a table of antilogarithms. Biirgi's more gen- 
eral point of view should also be mentioned. He de- 
sired to simplify all calculations by means of loga- 
rithms while Napier used only the logarithms of the 
trigonometric functions. 

Burgi was led to this method of procedure by 
comparison of the two series 0, 1, 2, 3, . . . and 1, 2, 
4, 8, ... or 20, 21, 22, 2», . . . He observed that for 
purposes of calculation it was most convenient to se- 
lect 10 as the base of the second series, and from this 
standpoint he computed the logarithms of ordinary 
numbers, though he first decided on publication when 
Napier's renown began to spread in Germany by rea- 
son of Kepler's favorable reports. Burgi's Geometric 
sche Progress Tabulen appeared at Prague in 1620,* 
and contained the logarithms of numbers from 10® to 
10* by tens. Burgi did not use the term logarithmus, 
but by reason of the way in which they were printed 
he called the logarithms **red numbers," the numbers 
corresponding, ** black numbers." 

Napier started with the observation that if in a 
circle with two perpendicular radii OA^ and OAi 
(r=:l), the sine ^o^i || OAi moves from O to A^ at 
intervals forming an arithmetic progression, its value 
decreases in geometrical progression. The segment 
OS^, Napier originally called numerus artificialis and 
later the direction number or logarithmus. The first 



publication of this new method of calculation, in which 
r = 10^ log sin 600 = 0, logsinO<>= 00, so that the log- 
arithms increased as the sines decreased, appeared in 
1614 and produced a great sensation. Henry Briggs 
had studied Napier's work thoroughly and made the 
important observation that it would be more suitable 
for computation if the logarithms were allowed to in^ 
crease with the numbers. He proposed to put log 1 
= 0, log 10 = 1, and Napier gave his assent. The, ta- 
bles of logarithms calculated by Briggs, on the basis 
of this proposed change, for the natural numbers from 
1 to 20 000 and from 90 000 to 100 000 were reckoned 
to 14 decimal places. The remaining gap was filled by 
the Dutch bookseller Adrian Vlacq. His tables which 
appeared in the year 1628 contained the logarithms of 
numbers from 1 to 100 000 to 10 decimal places. In 
these tables, under the name of his friend De Decker, 
Vlacq introduced logarithms upon the continent. As- 
sisted by Vlacq and Gellibrand, Briggs computed a 
table of sines to fourteen places and a table of tan- 
gents and secants to ten places, at intervals of 36". 
These tables appeared in 1633. Towards the close of 
the seventeenth century Claas Vooght published a 
table of sines, tangents, and secants with their loga- 
rithms, and, what was especially remarkable, they 
were engraved on copper. 

Thus was produced a collection of tables for logarithmic com- 
putation valuable for all time. This was extended by the intro- 
duction of the addition and subtraction logarithms always named 


after Gauss, but whose inventor, according to Gauss's own testi- 
mony, is Leonelli. The latter had proposed calculating a table 
with fourteen decimals ; Gauss thought this impracticable, and 
calculated for his own use a table with five decimals.* 

In the year 1875 there were in existence 553 logarithmic tables 
with decimal places ranging in number from 8 to 102. Arranged 
according to frequency, the 7-place tables stand at the head, then 
follow those with 5 places, 6-places, 4places, and 10-places. The 
only table with 102 places is found in a work by H. M. Parkhurst 
{Astronomical Tables, New York, 1871). 

Investigations of the errors occurring in logarithmic tables 
have been made by J. W. L. Glaisher.f It was there shown that 
every complete table had been transcribed, directly or indirectly 
after a more or less careful revision, from the table published in 
1628 which contains the results of Briggs's Arithmeiica logarith- 
mica of 1624 for numbers from 1 to 100000 to ten places. In the 
first seven places Glaisher found 171 errors of which 48 occur in 


the interval from 1 to 10000. These errors, due to Vlacq, have 
gradually disappeared. Of the mistakes in Vlacq, 98 still appear 
in Newton (1658), 19 in Gardiner (1742), 5 in Vega (1797), 2 in 
Callet (1855), 2 in Sang (1871). Of the tables tested by Glaisher, 
four turned out to be free from error, viz., those of Bremiker 
(1857), SchrSn (1860), Callet (1862), and Bruhns (1870). Contribu- 
tions to the rapid calculation of common logarithms have been 
made by Koralek (1861) and R. Hoppe (1876) ; the work of the 
latter is based upon the theorem that every positive number may 
be transformed into an infinite product. % 

*Gau88, Werke, III., p. 344. Porro in Bone. BuU., XVIII. 

\ Fortschritte, 1873. 

X Stolz, VorUsungen Uber allgemeine Aritkmetik, 1885-1886. 





After Regiomontanus had laid the foundations of 
plane and spherical trigonometry, and his successors 
had made easier the work of computation by the com- 
putation of the numerical values of the trigonomet- 
ric functions and the creation of a serviceable sys- 
tem of logarithms, the inner structure of the science 
was ready to be improved in details during this third 
period. Important innovations are especially due to 
Euler, who derived the whole of spherical trigonom- 
etry from a few simple theorems. Euler defined the 
trigonometric functions as mere numbers, so as to be 
able to substitute them for series in whose terms ap- 
pear arcs of circles from which the trigonometric func- 
tions proceed according to definite laws. From him 
we have a number of trigonometric formulae, in part 
entirely new, and in part perfected in expression. 
These were made especially clear when Euler denoted 
the elements of the triangle by a, b, c, a, ^, y. Then 
such expressions as sin a, tana could be introduced 
where formerly special letters had been used for the 
same purpose.* Lagrange and Gauss restricted them- 
selves to a single theorem in the derivation of spheri- 
cal trigonometry. The system of equations 

♦Baltzer, R., Die EUtnente der Maihevtatik. 1885. 


8,n - sm -y = Sin -^ cos ^^-g-^, 

with the corresponding relations, is ordinarily ascribed 
to Gauss, though the equations were first published 
by Delambre in 1807 (by Mollweide 1808, by Gauss 
1809).* The case of the Pothenot problem is similar: 
it was discussed by Snellius 1614, by Pothenot 1692, 
by Lambert 1766.t 

The principal theorems of polygonometry and 
polyhedrometry were established in the eighteenth 
century. To Euler we owe the theorem on the area 
of the orthogonal projection of a plane figure upon 
another plane ; to Lexell the theorem upon the pro- 
jection of a polygonal line. Lagrange, Legendre, 
Carnot and others stated trigonometric theorems for 
polyhedra (especially the tetrahedra), Gauss for the 
spherical quadrilateral. 

The nineteenth century has given to trigonometry 
a series of new formulae, the so-called projective for- 
mulae. Besides Poncelet, Steiner, and Gudermann, 
Mobius deserves special mention for having devised 
a generalization of spherical trigonometry, such that 
sides or angles of a triangle may exceed 180°. The im- 
portant improvements which in modern times trigono- 
metric developments have contributed to other mathe- 
matical sciences, may be indicated in this one sentence: 
their extended description would considerably en- 
croach upon the province of other branches of science. 

♦ Hammer, Lehrbuch der ebenen und sph'drischen Trigonometries 1897. 
tBaltzer, R., Die Elemente der Mathematik, 1885. 


Abel, Niels Henrik. Born at Fmd5e, Norway, August 5, 1802 ; 
died April 6, 1829. Studied in Christiania, and for a short 
time in Berlin and Paris. Proved the impossibility of the 
algebraic solution of the quintic equation ; elaborated the the- 
ory of elliptic functions ; founded the theory of Abelian func- 

Abui /ud, Mohammed ibn al Lait al Shanni. Lived about 1050. 
Devoted much attention to geometric problems not soluble 
with compasses and straight edge alone. 

Abul JVafa al Buzjani. Born at Buzjan, Persia, June 10, 940; 
died at Bagdad, July i, 998. Arab astronomer. Translated 
works of several Greek mathematicians ; improved trigonom- 
etry and computed some tables ; interested in geometric con- 
structions requiring a single opening of the compasses. 

Adelard. About 1120. English monk who journeyed through Asia 
Minor, Spain, Egypt, and Arabia. Made the first translation 
of Euclid from Arabic into Latin. Translated part of Al 
Khowarazmi's works. 

A I Battani (Albategnius). Mohammed ibn Jabir ibn Sinan Abu 
Abdallah al Battani. Bom in Battan, Mesopotamia, c. 850 ; 
died in Damascus, 929. Arab prince, governor of Syria ; great- 

*The translators feel that these notes will be of greater value to the 
reader by being arranged alphabetically than, as in the original, by periodt, 
especially as this latter arrangement is already given in the body of the 
work. They also feel that they will make the book more serviceable by 
changing the notes as set forth in the original, occasionally eliminating mat- 
ter of little consequence, and frequently adding to the meagre information 
given. They have, for this purpose, freely used such standard works as Can- 
tor, Hankel, Gtinther, Zeuthen, et al., and especially the valuable little Ztit- 
taftln zur Geschichte der Mathematik, Physik und Astronomie bis zum Jahre 
1500, by Felix Miiller, Leipzig, 1892. Dates are A. D., except when prefixed 
by the negative sign. 



est Arab astronomer and mathematician. Improved trigonom- 
etry and computed the first table of cotangents. 

AJberti, Leo Battista. 1404-1472. Architect, painter, sculptor. 

Albertus Magnus. Count Albrecht von BollstSldt. Born at Lan- 
ingen in Bavaria, 1193 or 1205; died at Cologne, Nov. 15, 
1280. Celebrated theologian, chemist, physicist, and mathe- 

M Btruni, Abul Rihan Mohammed ibn Ahmed. From Birun, 
valley of the Indus ; died 1038 Arab, but lived and travelled 
in India and wrote on Hindu mathematics. Promoted spheri- 
cal trigonometry. 

Alcuin. Born at York, 736; died at Hersfeld, Hesse, May 19 
804. At first a teacher in the cloister school at York ; then 
assisted Charlemagne in his efforts to establish schools in 

Alkazen, Ibn al Haitam. Bom at Bassora, 950 ; died at Cairo, 
1038. The most important Arab writer on optics. 

A I Kalsadiy Abul Hasan AH ibn Mohammed. Died i486 or 1477. 
From Andalusia or Granada. Arithmetician. 

Al Karkhi^ Abu Bekr Mohammed ibn al Hosain. Lived about 
loio. Arab mathematician at Bagdad. Wrote on arithmetic, 
algebra and geometry. 

Al Khojandi^ Abu Mohammed. From Khojand, in Khorassan ; 
was living in 992. Arab astronomer 

Al Khoivarazmi, Abu Jafar Mohammed ibn Musa. First part of 
ninth century. Native of Khwarazm (Khiva). Arab mathe- 
matician and astronomer. The title of his work gave the name 
to algebra. Translated certain Greek works. 

Al Kindi, Jacob ibn Ishak, Abu Yusuf . Born c. 813; died 873. 
Arab philosopher, physician, astronomer and astrologer. 

Al Kuhi, Vaijan ibn Rustam Abu Sahl. Lived about 975. Arab 
astronomer and geometrician at Bagdad. 

Al Nasaivi, Abul Hasan Ali ibn Ahmed. Lived about 1000 
From Nasa in Khorassan. Arithmetician. 

Al Sagani. Ahmed ibn Mohammed al Sagani Abu Hamid al Us- 
turlabi. From Sagan, Khorassan ; died 990. Bagdad astron- 


Anaxagoras. Bom at Clazomene, Ionia, — 499 ; died at Lamp- 
sacus, — 428. Last and most famous philosopher of the Ionian 
school. Taught at Athens. Teacher of Euripides and Pe- 

A planus (Apian), Petrus. Bom at Leisnig, Saxony, 1495 ; died in 
1552. Wrote on arithmetic and trigonometry. 

AfoUonius of Perga, in Pamphylia. Taught at Alexandria be- 
tween — 250 and — 200, in the reign of Ptolemy Philopator. 
His eight books on conies gave him the name of " the great 
geometer." Wrote numerous other works. Solved the gene- 
ral quadratic with the help of conies. 

Arbogast, Louis Fran9ois Antoine. Born at Mutzig, 1759 ; died 
1803. Writer on calculus of derivations, series, gamma func- 
tion, differential equations. 

Archimedes. Born at Syracuse, — 287(7) ; killed there by Roman 
soldiers in — 212. Engineer, architect, geometer, physicist. 
Spent some time in Spain and Egypt. Friend of King Hiero. 
Greatly developed the knowledge of mensuration of geometric 
solids and of certain curvilinear areas. In physics he is known 
for his work in center of gravity, levers, pulley and screw, 
specific gravity, etc. 

Archytas. Born at Tarentum — 430; died — 365. Friend of Plato, 
a Pythagorean, a statesman and a general. Wrote on propor- 
tion, rational and irrational numbers, tore surfaces and sec- 
tions, and mechanics. 

Argand, Jean Robert. Bom at Geneva, 1768 ; died c. 1825. Pri- 
vate life unknown. One of the inventors of the present method 
of geometrically representing complex numbers (1806). 

Aristotle. Born at Stageira, Macedonia, — 384 ; died at Chalcis, 
Euboea, —322. Founder of the peripatetic school of philoso- 
phy ; teacher of Alexander the Great. Represented unknown 
quantities by letters ; distinguished between geometry and 
geodesy ; wrote on physics ; suggested the theory of combina- 

Aryahhatta. Bom at Pataliputra on the Upper Ganges, 476. 
Hindu mathematician. Wrote chiefly on algebra, including 
quadratic equations, permutations, indeterminate equations, 
and magic squares. 


Auffust, Ernst Ferdinand. Bom at Prenzlau, 1795 ; died 1870 as 
director of the K6lnisch Realgymnasium in Berlin. 

Autolykus of Pitane, Asia Minor. Lived about — 330. Greek 
astronomer ; author of the oldest work on spherics. 

Avicenna. Abu AH Hosain ibn Sina. Bom at Char matin, neat 
Bokhara, 978 ; died at Hamadam, in Persia, 1036. Arab phy- 
sician and naturalist. Edited several mathematical and phys- 
ical works of Aristotle, Euclid, etc. Wrote on arithmetic and 

Babbage, Charles. Born at Totnes, Dec. 26, 1792 ; died at Lon- 
don, Oct. 18, 1871. Lucasian professor of mathematics at 
Cambridge. Popularly known for his calculating machine. 
Did much to raise the standard of mathematics in England. 

Bachet. See M^ziriac. 

Btuon, Roger. Born at Ilchester, Somersetshire. 1214; died at 
Oxford, June 11, 1294. Studied at Oxford and Paris; profes- 
sor at Oxford ; mathematician and physicist. 

BaUms. Lived about 100. Roman surveyor. 

Baldi, Bernardino. Born at Urbino, 1553; died there, 1617. 
Mathematician and general scholar. Contributed to the his- 
tory of mathematics. 

BaUzer, Heinrich Richard. Born at Meissen in 1818; died at 
Giessen in 1887. Professor of mathematics at Giessen. 

Barlaam, Bernard Beginning of fourteenth century. A monk 
who wrote on astronomy and geometry. 

Barozzi, Francesco. Italian mathematician. 1 537-1604. 

Barrozu, Isaac. Born at London, 1630; died at Cambridge, May 
4, 1677. Professor of Greek and mathematics at Cambridge. 
Scholar, mathematician, scientist, preacher, llewton was his 
pupil and successor. 

Beda, the Venerable. Born at Monkton, near Yarrow, Northum- 
berland, in 672; died at Yarrow, May 26, 735. Wrote on chro- 
nology and arithmetic. 

Bellavitis, Giusto. Born at Bassano, near Padua, Nov. 22, 1803; 
died Nov. 6, 1880. Known for his work in projective geom 
etry and his method of equipollences. 


Bernelinus. Lived about 1020. Pupil of Gerbert at Paris. Wrote 
on arithmetic. 

Bernoulli. Famous mathematical family. 

Jacob (often called James, by the English), bom at Basel, Dec. 
27. 1654 ; died there Aug. 16, 1705. Among the first to recog- 
nize the value of the calculus. His De Arte Conj'ectemdi is a 
classic on probabilities. Prominent in the study of curves, the 
logarithmic spiral being engraved on his monument at Basel. 

yohn (Johann), his brother ; bom at Basel, Aug. 7, 1667 ; died 
there Jan. i, 1748. Made the first attempt to construct an 
integral and an exponential calculus. Also prominent as a 
physicist, but his abilities were chiefly as a teacher. 

Nicholas (Nikolaus), his nephew ; born at Basel, Oct. 10, 1687 ; 
died there Nov. 29, 1759. Professor at St. Petersburg, Basel, 
and Padua. Contributed to the study of differential equations. 

Daniel, son of John ; born at GrOningen, Feb. g, 1700 ; died at 
Basel in 1782. Professor of mathematics at St. Petersburg. 
His chief work was on hydrodynamics. 

John the younger, son of John. 17 10- 1790. Professor at Basel. 

Bizout, i^tienne. Born at Nemours in 1730 ; died at Paris in 
1783. Algebraist, prominent in the study of symmetric func- 
tions and determinants. 

Bhaskara Acharya. Born in 11 14. Hindu mathematician and 
astronomer. Author of the LHavati and the Vijaganitat con- 
taining the elements of arithmetic and algebra. One of the 
most prominent mathematicians of his time. 

Biott Jean Baptiste. Born at Paris, Apr. 21, 1774; died same 
place Feb. 3, 1862. Professor of physics, mathematics, as- 
tronomy. Voluminous writer. 

Boeihius, Anicius Manlius Torquatus Severinns. Bom at Rome, 
480 ; executed at Pavia, 524. Founder of medieval scholasti 
cism. Translated and revised many Greek writings on math- 
ematics, mechanics, and physics. Wrote on arithmetic. While 
in prison he composed his Consolations of Philosophy. 

Bolyai: Wolfgang Bolyai de Bolya. Born at Bolya, 1775 ; died 
in 1856. Friend of Gauss. 

Johann Bolyai de Bolya, his son. Born at Klausenburg, 1802 ; 
died at Maros-Vasarhely, i860. One of the discoverers (see 
Lobachevsky) of the so-called non- Euclidean geometry. 


Boixano, Bemhard. 1781-1848. Contribated to the stndy of 

BombeUi, Rafaele. Italian. Bom c. 1530. His algebra (1572) 
summarized all then known on the subject. Contribated to 
the study of the cubic. 

Boncom^gni, Baldassare. Wealthy Italian prince. Bom at Rome. 
May 10, 1821 ; died at same place, April 12, 1894. Publisher 
of Boncompagni's Bulletino. 

Boole, George. Bom at Lincoln. 1815 ; died at Cork, 1864. Pro- 
fessor of mathematics in Queen's College, Cork. The theory 
of invariants and covariants may be said to start with his con- 
tributions (1841). 

Booth, James. 1806-1878. Clergyman and writer on elliptic in- 

Borchardt, Karl Wilhelm. Bom in 1817 ; died at Berlin, 1880. 
Professor at Berlin. 

Boschi, Pietro. Bom at Rome. 1833 ; died in 1887. Professor 
at Bologna. 

Bouquet, Jean Claude. Bom at Morteau in 1819 ; died at Paris. 

Bour, Jacques Edmond i^mile. Bom in 1832 ; died at Paris, 1866. 
Professor in the ^cole Polytechnique. 

Bradtvardine, Thomas de. Born at Hard field, near Chichester, 
1290; died at Lambeth. Aug. 26, 1349. Professor of theology 
at Oxford and later Archbishop of Canterbury. Wrote upon 
arithmetic and geometry. 

Brahmagufta. Born in 598. Hindu mathematician. Contrib 
uted to geometry and trigonometry. 

Brasseur, Jean Baptiste. 1802-1868. Professor at Liege. 

Bretschneider, Carl Anton. Born at Schneeberg, May 27, i8o8 . 
died at Gotha, November 6, 1878. 

Brianchon, Charles Julien. Born at Sevres, 1785 ; died in 1864. 
Celebrated for his reciprocal (1806) to Pascal's mystic hexa- 

Briggs, Henry. Born at Warley Wood, near Halifax, Yorkshire, 
Feb. 1560-1 ; died at Oxford Jan. 26, 1630-1. Savilian Pro- 


fessor of geometry at Oxford. Among the first to recognize 
the value of logarithms; those with decimal base bear his 

Briot, Charles August Albert. Bom at Sainte-Hippolyte, 18 17; 
died in 1882. Professor at the Sorbonne, Paris. 

Brouncker, William, Lord. Born in 1620 (?) ; died at Westminster, 
1684. First president of the Royal Society. Contributed to 
the theory of series. 

Brunelleschi, Filippo. Born at Florence, 1379; died there April 
16, 1446. Noted Italian architect. 

BUrgi, Joost (Jobst). Born at Lichtensteig, St. Gall, Switzerland, 
1552 ; died at Cassel in 1632. One of the first to suggest a 
system of logarithms. The first to recognize the value of mak- 
ing the second member of an equation zero. 

Ca^orali^ Bttore. Born at Perugia, 1855 ; died at Naples, 1886. 
Professor of mathematics and writer on geometry. 

Cardan^ Jerome (Hieronymus, Girolamo). Born at Pavia, 1501 ; 
died at Rome, 1576. Professor of mathematics at Bologna and 
Padua. Mathematician, physician, astrologer. Chief contri- 
butions to algebra and theory of epicycloids. 

Carnot, Lazare Nicolas Marguerite. Born at Nolay, C6te d'Or, 
1753 ; died in exile at Magdeburg, 1823. Contributed to mod- 
ern geometry. 

Casstni, Giovanni Domenico. Born at Perinaldo, near Nice, 1625; 
died at Paris, 17 12. Professor of astronomy at Bologna, and 
first of the family which for four generations held the post of 
director of the observatory at Paris. 

Castig-lianOt Carlo Alberto. 1847- 1884. Italian engineer. 

Catalan, Eugene Charlps. Born at Bruges, Belgium, May 30, 
1814 ; diad Feb. 14, 1894. Professor of mathematics at Paris 
and Liege. 

Cataldi, Pietro Antonio. Italian mathematician, bom 1548 ; died 
at Bologna, 1626. Professor of mathematics at Florence, 
Perugia and Bologna. Pioneer in the use of continued frac- 

Cattaneo, Francesco. 1811-1875. Professor of physics and me- 
chanics in the University of Pavia. 


Cauchy, Angastin Louis. Bom at Paris, 1789 ; died at Sceanz, 
1857. Professor of mathematics at Paris. One of the most 
prominent mathematicians of his time. Contributed to the 
theory of functions, determinants, differential equations, the- 
ory of residues, elliptic functions, convergent series, etc. 

Cavaiieri, Bonaventura. Born at Milan, 1598 ; died at Bologna, 
1647. Paved the way for the differential calculus by his 
method of indivisibles (1629). 

Cayley, Arthur Born at Richmond, Surrey, Aug. 16, 1821 ; died 
at Cambridge, Jan. 26, 1895. Sadlerian professor of mathe- 
matics. University of Cambridge. Prolific writer on mathe- 

Ceva, Giovanni. 1648-^. 1737. Contributed to the theory of trans- 

Chasles, Michel. Born at Chartres, Nov. 15, 1793 ; died at Paris, 
Dec. 12, 1880. Contributed extensively to the theory of mod- 
em geometry. 

Chelini, Domenico. Born 1802; died Nov. 16, 1878. Italian mathe- 
matician ; contributed to analytic geometry and mechanics. 

Chuquet, Nicolas. From Lyons ; died about 1500. Lived in Paris 
and contributed to algebra and arithmetic. 

Clairaut, Alexis Claude. Born at Paris, 1713 ; died there, 1765. 
Physicist, astronomer, mathematician. Prominent in the study 
of curves. 

Clausberg-, Christlieb von. Born at Danzig, 1689 ; died at Copen- 
hagen, 175 1. 

Clehsch, Rudolf Friedrich Alfred. Bom January 19, 1833 ; died 
Nov. 7, 1872. Professor of mathematics at Carlsruhe, Giessen 
and GOttingen. 

Condorcet, Marie Jean Antoine Nicolas. Born at Ribemont, near 
St. Quentin, Aisne, 1743 ; died at Bourg-la Reine, 1794. Sec- 
retary of the Academic des Sciences. Contributed to the the- 
ory of probabilities. 

Cotes, Roger. Bom at Burbage, near Leicester, July 10, 1682 ; 
died at Cambridge, June 5, 1716. Professor of astronomy at 
Cambridge. His name attaches to a number of theorems in 
geometry, algebra and analysis. Newton remarked, "If Cotes 
had lived we should have learnt something." 


Cramer, Gabriel. Bom at Geneva, 1704 ; died at Bagnols, 1752. 
Added to the theory of equations and revived the study of de- 
terminants (begun by Leibnitz). Wrote a treatise on curves. 

Crelle, August Leopold. Born at Eichwerder (Wriezen a. d. Oder), 
1780 ; died in 1855. Founder of the Journal fUr reine und 
angewandte Mathematik (1826). 

D'Alembert, Jean le Rond. Born at Paris, 1717 ; died there, 1783 
Physicist, mathematician, astronomer. Contributed to the 
theory of equations. 

De Beaune, Florimond. 1601-1652. Commentator on Descartes's 

De la Gournerie, Jules Antoine Ren^ Maillard. Born in 1814 ; 
died at Paris, 1883. Contributed to descriptive geometry. 

Del Monte^ Guidobaldo. 1 545-1607. Wrote on mechanics and 

Democritus. Bom at Abdera, Thrace, — 460 ; died c. — 370. Stud- 
ied in Egypt and Persia. Wrote on the theory of numbers and 
on geometry. Suggested the idea of the infinitesimal. 

De Moivre, Abraham. Born at Vitry, Champagne, 1667 ; died at 
London, 1754. Contributed to the theory of complex num- 
bers and of probabilities 

De Morgan, Augustus. Bom at Madura, Madras, June 1806 ; 
died March 18, 1871. First professor of mathematics in Uni- 
versity of llondon (1828) Celebrated teacher, but also con- 
tributed to algebra and the theory of probabilities. 

Desargues, Gerard. Born at Lyons, 1593 ; died in 1662. One of 
the founders of modern geometry. 

Descartes, Rene, du Perron. Born at La Haye, Touraine, 1596; 
died at Stockholm, 1650. Discoverer of analytic geometry. 
Contributed extensively to algebra. 

Dinostratus. Lived about — 335. Greek geometer. Brother of 
Menaechmus. His name is connected with the quadratrix. 

Diodes. Lived about — 180. Greek geometer. Discovered the 
cissoid which he used in solving the Delian problem. 

Diofhantus of Alexandria. Lived about 275. Most prominent of 
Greek algebraists, contributing especially to indeterminate 



Dirichlet, Peter Gnstav Lejeune. Born at Dtiren. 1805 ; died at 
Gdttingen, 1859. Succeeded Gauss as professor at GOttingen. 
Prominent contributor to the theory of numbers. 

Dodson, James. Died Nov. 23, 1757. Great grandfather of De 
Morgan. Known chiefly for his extensive table of anti-log- 
arithms (1742). 

DonateUo, 1386-1468. Italian sculptor. 

Du Bois-Reymond, Paul David Gustav. Born at Berlin, Dec. 2, 
1831 ; died at Freiburg, April 7, 1889. Professor of mathe- 
matics in Heidelberg, Freiburg, and Ttibingen. 

DuTiamel, Jean Marie Constant. Born at Saint-Malo, 1797 ; died 
at Paris, 1872. One of the first to write upon method in math- 

Dufin, Franyois Pierre Charles. Born at Varzy, 1784 ; died at 
Paris, 1873. 

DUrer\ Albrecht. Born at Nuremberg, 1471 ; died there, 1528. 
Famous artist. One of the founders of the modern theory of 

Eisenstein, Ferdinand Gotthold Max. Bom at Berlin, 1823 ; died 
there, 1852. One of the earliest workers in the field of invari- 
ants and covariants. 

Enne^er^ Alfred. 1830-1885. Professor at GOttingen. 

Ej>a^hroditus. Lived about 200. Roman surveyor. Wrote on 
surveying, theory of numbers, and mensuration. 

Eratosthenes. Born at Cyrene, Africa, — 276 ; died at Alexan- 
dria, — 194. Prominent geographer. Known for his " sieve " 
for finding primes. 

Euclid. Lived about — 300. Taught at Alexandria in the reign 
of Ptolemy Soter. The author or compiler of the most famous 
text-book of Geometry ever written, the Elements, in thirteen 

Eudoxus of Cnidus. — 408, — 355. Pupil of Archytas and Plato. 
Prominent geometer, contributing especially to the theories of 
proportion, similarity, and "the golden section." 

Euler, Leonhard. Born at Basel. 1707 ; died at St. Petersburg, 
1783. One of the greatest physicists, astronomers and math- 
ematicians of the i8th century. "In his voluminous . . . 


writings will be found a perfect storehouse of investigations 
on every branch of algebraical and mechanical science." — 

Eutodus. Bom at Ascalon, 480. Geometer. Wrote commen- 
taries on the works of Archimedes, Apollonius, and Ptolemy. 

Fagnano, GiulioCarlo, Count de. Bom at Sinigaglia, 1682 ; died 
in 1766. Contributed to the study of curves. Euler credits 
him with the first work in elliptic functions. 

Faulhaher, Johann. 1580-1635. Contributed to the theory of 

Fermal, Pierre de. Born at Beaumont-de-Lomagne, near Mon- 
tauban, 1601 ; died at Castres, Jan. 12, 1665. One of the most 
versatile mathematicians of his time ; his work on the theory 
of numbers has never been equalled. 

Ferrari, Ludovico. Born at Bologna, 1522 ; died in 1562. Solved 
the biquadratic. 

Ferro, Scipione del. Born at Bologna, c. 1465 ; died between 
Oct. 29 and Nov. 16, 1526. Professor of mathematics at Bo- 
logna. Investigated the geometry based on a single setting of 
the compasses, and was the first to solve the special cubic 

Feuerbac/i, Karl Wilhelm Born at Jena, 1800 ; died in 1834. 
Contributed to modern elementary geometry. 

Fibonacci. See Leonardo of Pisa. 

Fourier, Jean Baptiste Joseph, Baron. Born at Auzerre, 1768 ; 
died at Paris, 1830. Physicist and mathematician. Contrib- 
uted to the theories of equations and of series. 

Frinicle. Bernard FrSnicle de Bessy. 1605-1675. Friend of 


Frlzier, Am^dee Franjois. Born at Chamb^ry, 1682 ; died at 
Brest, 1773. One of the founders of descriptive geometry. 

Friedlein, Johann Gottfried. Born at Regensburg, 1828 ; died in 

Frontinus, Sextus Julius. 40-103. Roman surveyor and engineer. 

Galois, Evariste. Born at Paris, 181 1 ; died there, 1832. Founder 
of the theory of groups. 


Gauss, Karl Friedrich. Bom at Brunswick, 1777; died at GOt- 
tingen, 1855. '^^^ greatest mathematician of modern times. 
Prominent as a physicist and astronomer. The theories of 
numbers, of functions, of equations, of determinants, of com- 
plex numbers, of hyperbolic geometry, are all largely indebted 
to his great genius. 

Geber. Jabir ben Aflah. Lived about 1085. Astronomer at Se- 
ville ; wrote on spherical trigonometry. 

Gelitbrand, Henry. 1597-1637. Professor of astronomy at Gresham 

Gemtnus. Bom at Rhodes, — 100 ; died at Rome, — 40. Wrote 
on astronomy and (probably) on the history of pre-Euclidean 

Gerbert, Pope Sylvester II. Bom at Auvergne, 940 ; died at 
Rome, May 13, 1003. Celebrated teacher; elected pope in 
999. Wrote upon arithmetic. 

Gerhard of Cremona. From Cremona (or, according to others, 
Carmona in Andalusia). Born in 11 14; died at Toledo in 
1187. Physician, mathematician, and astrologer. Translated 
several works of the Greek and Arab mathematicians from 
Arabic into Latin. 

Germain, Sophie. 1776 183 1. Wrote on elastic surfaces. 

Girard, Albert, c. 1590-1633. Contributed to the theory of equa- 
tions, general polygons, and symbolism. 

Gb'pel, Gustav Adolf. 1812-1847. Known for his researches on 
hyperelliptic functions. 

Grammateus, Henricus. (German name, Heinrich Schreiber.) 
Bom at Erfurt, c. 1476. Arithmetician. 

Grassmann, Hermann Gtinther. Born at Stettin, April 15. 1809; 
died there Sept. 26, 1877. Chiefly known for his Ausde/inungs- 
lehre (1844). Also wrote on arithmetic, trigonometry, and 

Grebe, Ernst Wilhelm. Born near Marbach, Oberhesse, Aug. 30, 
1804 > ^^^^ ^^ Cassel, Jan. 14, 1874. Contributed to modern 
elementary geometry. 

Gregory, James. Born at Drumoak, Aberdeenshire, Nov. 1638 ; 
died at Edinburgh, 1675. Professor of mathematics at St. An- 


drews and Edinburgh. Proved the incommensurabilitj of ir ; 
contributed to the theory of series. 

Grunert, Johann August. Bom at Halle a. S., 1797; died in 1872 
Professor at Greifswalde, and editor of Grunert's Archiv. 

Gua. Jean Paul de Gua de Malves. Bom at Carcassonne, 171 3 : 
died at Paris, June 2, 1785. Gave the first rigid proof of Des 
cartes's rule of signs. 

Gudermann, Christoph. Bom at Winnebnrg, March 28, 1798 ; 
died at Monster, Sept. 25, 1852. To him is largely due the 
introduction of hyperbolic functions into modern analysis. 

Guldtn, Habakkuk (Paul). Born at St. Gall, 1577; died at GrStz, 
1643. Known chiefly for his theorem on a solid of revolution, 
pilfered from Pappus. 

Hachette, Jean Nicolas Pierre. Born at M^zieres, 1769 ; died at 
Paris, 1834. Algebraist and geometer. 

Halley, Edmund. Born at Haggerston, near London. Nov. 8, 
1656 : died at Greenwich, Jan. 14, 1742. Chiefly known for 
his valuable contributions to physics and astronomy. 

Halfhen, George Henri. Born at Rouen, Oct. 30, 1844 ; died at 
Versailles in 1889. Professor in the fecole Polytechnique at 
Paris. Contributed to the theories of differential equations 
and of elliptic functions. 

Hamilton, Sir William Rowan. Bom at Dublin, Aug. 3-4, 1805 ; 
died there, Sept. 2, 1865. Professor of astronomy at Dublin. 
Contributed extensively to the theory of light and to dynamics, 
but known generally for his discovery of quaternions. 

flankel, Hermann. Born at Halle, Feb. 14, 1839 ; died at Schram- 
berg, Aug. 29, 1873. Contributed chiefly to the theory of com- 
plex numbers and to the history of mathematics. 

Harnack, Karl Gustav Axel. Born at Dorpat, 1851; died at Dres- 
den in 1888. Professor in the polytechnic school at Dresden. 

Harriot, Thomas. Born at Oxford, 1560; died at Sion House, 
near Isle worth, July 2, 162 1. The most celebrated English 
algebraist of his time. 

^^r<>« of Alexandria. Lived about — no. Celebrated surveyor 
and mechanician. Contributed to mensuration. 


Hesse, Lod wig Otto. Bom at K6nigsberg, April 22, 181 1 ; died 
at Munich. Aug. 4. 1874. Contributed to the theories of curves 
and of determinants. 

ffipparchus. Bom at Nicaea, Bithynia, — 180 ; died at Rhodes, 
— 225. Celebrated astronomer. One of the earliest writers 
on spherical trigonometry. 

Hifpias of Elis. Bom c. — 460. Mathematician, astronomer, 
natural scientist. Discovered the quadratrix. 

Hippocrates of Chios. Lived about — 440. Wrote the first Greek 
elementary text-book on mathematics. 

Homer, William George. Born in 1786 ; died at Bath, Sept. 22, 
1837. Chiefly known for his method of approximating the real 
roots of a numerical equation (18 19). 

Hrabanus Maurus. 788-856. Teacher of mathematics. Arch 
bishop of Mainz. 

Hudde, ]ohann. Born at Amsterdam, 1633 ; died there, 1704. 
Contributed to the theories of equations and of series. 

Honein ibn Ishak. Died in 873. Arab physician. Translated 
several Greek scientific works. 

Huygens, Christiaan, van Zuylichem. Born at the Hag^e, 1629 ; 
died there, 1695. Famous physicist and astronomer. In math- 
ematics he contributed to the study of curves. 

Hyginus. Lived abont 100. Roman surveyor. 

HyfcUia, daughter of Theon of Alexandria. 375-415. Composed 
several mathematical works. See Charles Kingsley's Hyfatia. 

Hyf steles of Alexandria. Lived about — 190. Wrote on solid 
geometry and theory of numbers, and solved certain indeter- 
minate equations. 

famblichus. Lived about 325. From Chalcis. Wrote on various 
branches of mathematics. 

Ibn al Banna. Abul Abbas Ahmed ibn Mohammed ibn Otman al 
Azdi al Marrakushi ibn al Banna Algarnati. Born 1252 or 
1257 '^^ Morocco. West Arab algebraist ; prolific writer. 

/bn Yunus, Abul Hasan Ali ibn Abi Said Abderrahman. 960- 
1008. Arab astronomer ; prepared the Hakimitic Tables. 


Isidorus His^alensis. Born at Carthagena, 570 ; died at Seville, 
636. Bishop of Seville. His Origines contained dissertations 
on mathematics. 

Ivory ^ James. Bom at Dundee, 1765 ; died at London, Sept. 21, 
1842. Chiefly known as a physicist. 

Jacobi, Karl Gustav Jacob. Born at Potsdam, Dec. 10, 1804; 
died at Berlin, Feb. 18, 185 1. Important contributor to the 
theory of elliptic and theta functions and to that of functional 

Jamin, Jules C^lestin. Bom in 1818; died at Paris, 1886. Pro- 
fessor of physics. 

Joannes de Praga (Johannes Schindel). Born at K6niggr£ltz, 1370 
or 1375 ; died at Prag c. 1450 Astronomer and mathema- 

Johannes of Seville (Johannes von Luna, Johannes Hispalensis). 
Lived about 1140. A Spanish Jew; wrote on arithmetic and 

Johann von GmUnden, Bom at GmQnden am Traunsee, between 
1375 and 1385 ; died at Vienna, Feb. 23, 1442. Professor of 
mathematics and astronomy at Vienna ; the first full professor 
of mathematics in a Teutonic university. 

Kdstner, Abraham Gotthelf. Bom at Leipzig, 1719; died at GCt- 
tingen, 1800. Wrote on the history of mathematics. 

JCepler, Johann. Born in Wttrtemberg, near Stuttgart, 1571 ; died 
at Regensburg, 1630. Astronomer (assistant of Tycho Brahe, 
as a young man); "may be said to have constructed the edi- 
fice of the universe," — Proctor. Prominent in introducing the 
use of logarithms. Laid down the ' ' principle of continuity " 
(1604); helped to lay the foundation of the infinitesimal cal- 

KTiayyam, Omar. Died at Nishapur, 1123. Astronomer, geometer, 
algebraist. Popularly known for his famous collection of 
quatrains, the Rvbaiyat. 

Kdhel, Jacob. Bom at Heidelberg, 1470 ; died at Oppenheim, in 
1533. Prominent writer on arithmetic (1514. 1520). 

LacroiXt Sylvestre Fran9ois. Born at Paris, 1765; died there, 
May 25, 1843. Author of an elaborate course of mathematics. 


LcLguerre, Edmoni Nicolas. Born at Bar-le-Duc, April 9, 1834 ; 
died there Aug. 14, 1886. Contributed to higher analysis. 

Lagrange, Joseph Louis, Comte. Born at Turin, Jan. 25, 1736 ; 
died at Paris, April 10, 181 3. One of the foremost mathe- 
maticians of his time. Contributed extensively to the calculus 
of variations, theory of numbers, determinants, differential 
equations, calculus of finite differences, theory of equations, 
and elliptic functions. Author of the Micanique analytique. 
Also celebrated as an astronomer. 

Lahire^ Philippe de. Born at Paris, March 18, 1640; died there 
April 21, 1 7 18. Contributed to the study of curves and magic 

LcUoubere, Antoine de. Bom in Languedoc, 1600; died at Tou- 
louse, 1664. Contributed to the study of curves. 

Lambert, Johann Heinrich. Born at Miilhausen, Upper Alsace, 
1728; died at Berlin, 1777. Founder of the hyperbolic trigo- 

Lami, Gabriel. Bom at Tours, 1795 ; died at Paris, 1870. Writer 
on elasticit>. and orthogonal surfaces. 

Landen, John. Born at Peakirk, near Peterborough, 1719 ; died 
at Milton, 1790. A theorem of his (1755) suggested to Euler 
and Lagrange their study of elliptic integrals. 

Laflace, Pierre Simon, Marquis de. Bom at Beaumont-en- Auge, 
Normandy, March 23, 1749; died at Paris, March 5, 1827. 
Celebrated astronomer, physicist, and mathematician. Added 
to the theories of least squares, determinants, equations, se- 
ries, probabilities, and differential equations. 

Legendre, Adrien Marie. Born at Toulouse, Sept. 18, 1752 ; died 
at Paris, Jan. 10, 1833. Celebrated mathematician, contribut- 
ing especially to the theory of elliptic functions, theory of 
numbers, least squares, and geometry. Discovered the "la'v^ 
of quadratic reciprocity," — "the gem of arithmetic" (Gauss). 

Leibnitz, Gottfried Wilhelm. Born at Leipzig, 1646; died at 
Hanover in 17 16. One of the broadest scholars of modem 
times; equally eminent as a philosopher and mathematician. 
One of the discoverers of the infinitesimal calculus, and the 
inventor of its accepted symbolism. 


Leonardo of Pisa, Fibonacci (filins Bonacii, son of Bonacins). 
Born at Pisa, 1180; died c. 1250. Travelled extensively and 
brought back to Italy a knowledge of the Hindu numerals and 
the general learning of the Arabs, which he set forth in his 
Liber Abaci,- Practica geometriae, and Flos. 

L* Hospital, Guillaume Fran9ois Antoine de, Marquis de St. 
Mesme. Born at Paris, 1661 ; died there 1704. One of the 
first to recognise the value of the infinitesimal calculus. 

LhuiUer, Simon Antoine Jean. Born at Geneva, 1750; died in 
1840. Geometer. 

Libri, Carucci dalla Sommaja, Guglielmo Brutus Icilius Timoleon. 
Born at Florence, Jan. 2, 1803; died at Villa Fiesole, Sept. 
28, 1869 Wrote on the history of mathematics in Italy. 

Lie, Marius Sophus. Bom Dec. 12. 1842 ; died Feb. 18, 1899. 
Professor of mathematics in Christiania and Leipzig. Spe- 
cially celebrated for his theory of continuous groups of trans- 
formations as applied to dififerential equations. 

LiouviUe, Joseph. Bom at St. Omer, 1809 ; died in 1882. Founder 
of the journal that bears his name. 

Lobachevsky, Nicolai Ivanovich. Bom at Makarief, 1793; died 
at Kasan, Feb 12-24, 1856. One of the founders of the so- 
called non-Eucliflean geometry. 

Ludolfh van Ceulen, See Van Ceulen. 

MacCullagh, James. Born near Strabane, 1809 ; died at Dublin, 
1846. Professor of mathematics and physics in Trinity Col- 
lege, Dublin. _ 

Maclaurin, Colin. Born at Kilmodan, Argyllshire, 1698; died at 
York, June 14, 1746. Professor of mathematics at Edinburgh. 
Contributed to the study of conies and series 

Malfatti, Giovanni Francesco Giuseppe. Bom at Ala, Sept. 26, 
1731 ; died at Ferrara, Oct. 9, 1807. Known for the geomet- 
ric problem which bears his name, 

Mains, £tienne Louis. Born at Paris, June 23, 1775; died there, 
Feb. 24, 1812. Physicist. 

Mixsrheroni, Lorenzo. Born at Castagneta, 1750; died at Paris, 
f 800. First to elaborate the geometry of the compasses only 


Maurolico, Francesco. Born at Messina, Sept. x6, 1494; died 
July 21, 1575. The leading geometer of his time. Wrote also 
on trigonometry. 

Maximus Planudes. Lived about 1330. From Nicomedia. Greek 
mathematician at Constantinople. \ . rote a commentary on 
Diophantus ; also on arithmetic. 

Menaechmus. Lived about — 350. Pupil of Plato. Discoverer 
of the conic sections. 

Menelaus of Alexandria. Lived about 100. Greek mathematician 
and astronomer. Wrote on geometry and trigonometry. 

Mercator^ Gerhard. Born at Rupelmonde, Flanders, 1512 : died 
at Duisburg, 1594. Geographer. 

Mercator, Nicholas. (German name Kaufmann.) Bom near 
Cismar, Holstein, c. 1620; died at Paris, 1687. Discovered 
the series for log (1 -|-^)- 

Melius, Adriaan. Born at Alkmaar, 157 1 ; died at Franeker, 1635 
Suggested an approximation for ir, really due to his father. 

Meusnier de la Place, Jean Baptiste Marie Charles. Bom at 
Paris, 1754 ; died at Cassel, 1793. Contributed a theorem on 
the curvature of surfaces. 

M^ziriac, Claude Gaspard Bachet de. Born at Bourg-en-Bresse, 
1581 ; died in 1638. Known for his Problhmes plaisants, etc. 
(1624) and his translation of Diophantus. 

Mdbins, August Ferdinand. Born at Schulpforta, Nov. 17, 1790 ; 
died at Leipzig, Sept. 26, 1868. One of the leaders in modei:n 
geometry. Author of Der Barycentrische CalciU {iS2y). 

Mohammed ibn Musa. See Al Khowarazmi. 

Moivre. See DeMoivre. 

MoUweide, Karl Brandan. Born at Wolfenbflttel, Feb. 3, 1774 ; 
died at Leipzig, March 10, 1825. Wrote on astronomy and 

Monge, Gaspard, Comte de P^luse. Born at Beaune, 1746 ; died 
at Paris, 1818. Discoverer of descriptive geometry; contrib- 
uted to the study of curves and surfaces, and to dififerential 


Montmort, Pierre R^mond de. Bom at Paris, 1678 ; died there, 
1719. Contributed to the theory of probabilities and to the 
summation of series. 

Moschofulus, Manuel. Lived about 1300. Byzantine mathemati- 
cian. Known for his work on magic squares. 

Mydorge^ Claude. Bom at Paris, 1585 ; died there in 1647. Auihor 
of the first French treatise on conies. 

Napier, John. . Born at Merchiston, then a suburb of Edinburgh, 
1550 ; died there in 1617. Inventor of logarithms. Contrib- 
uted to trigonometry. 

Newton ^ Sir Tsaac. Bom at Woolsthorpe, Lincolnshire, Dec. 25, 
1642, O. S. ; died at Kensington, March 20, 1727. Succeeded 
Barrow as Lucasian professor of mathematics at Cambridge 
(1669). The world's greatest mathematical physicist. Invented 
fluxional calculus {c. 1666). Contributed extensively to the 
theories of series, equations, curves, and, in general, to all 
branches of mathematics then known. 

Nicole, Fran9ois. Born at Paris, 1683; died there, 1758. First 
treatise on finite dififerences. 

NicomcLchus of Gerasa, Arabia. Lived 100. Wrote upon arith- 

Nicomedes of Gerasa. Lived — 180. Discovered the conchoid 
which bears his name. 

Nicolaus von Cusa. Born at Cuss on the Mosel, 1401 ; died at 
Todi, Aug. II, 1464. Theologian, physicist, astronomer, ge- 

Odo of Cluny. Born at Tours, 879 ; died at Cluny, 942 or 943. 
Wrote on arithmetic. 

Oenopides of Chios. Lived — ^465. Studied in Egypt. Geometer. 

Olivier, Theodore. Born at Lyons, Jan. 21, 1793 : died in same 
place Aug. 5, 1853. Writer on descriptive geometry. 

Oresme, Nicole. Born in Normandy, c. 1320; died at Lisieux, 
1382. Wrote on arithmetic and geometry. 

Oughtred, William. Born at Eton, 1574 ; died at Albury, 1660. 
Writer on arithmetic and trigonometry. 

Pacioli, Luca. Fra Luca di Borgo di Santi Sepulchri. Bora at 
Borgo San Sepolcro, Tuscany, c. 1445 ; died at Florence, 


c. 1509. Taught in several Italian cities. His Summa de 
Arithmetica, Geomelria, etc., was the first great mathemat- 
ical work published (1494V 
Papfnis of Alexandria. Lived about 300. Compi'ed a work con 
taining the mathematical knowledge of his time. 

Parent, Antoine. Bom at Paris, 1666 ; died there in 1716. Fiist 
to refer a surface to three co-ordinate planes (1700). 

Pascal, Blaise. Born at Clermont, 1623; died at Paris, 1662 
Physicist, philosopher, mathematician. Contributed to the 
theory of numbers, probabilities, and geometry 

Peirce, Charles S. Bom at Cambridge, Mass., Sept. 10, 1839. 
Writer on logic. 

PeU, John. Born in Sussex, March 1, 1610 ; died at London, Dec 
10, 1685. Translated Rahn's algebra. 

Perseus. Lived — 150. Greek geometer ; studied spiric lines. 

Peuerbach, Georg von. Born at Peuerbach, Upper Austria, May 
30, 1423; died at Vienna, Aprils, 1461. Prominent teacher 
and writer on arithmetic, trigonometry, and astronomy. 

P/aff, Johann Friedrich. Born at Stuttgart, 1765 ; died at Halle 
in 1825 Astronomer and mathematician. 

Pitiscus, Bartholomaeus. Born Aug. 24, 1561 ; died at Heidel- 
berg, July 2, 1613. Wrote on trigonometry, and first used the 
present decimal point (1612). 

Plana, Giovanni Antonio Amedeo. Born at Voghera, Nov. 8. 
1781; died at Turin, Jan. 2, 1864. Mathennatical astronomer 
and physicist. 

Planudfs. See Maximus Planudes. 

f%/^aM, Joseph Antoine Ferdinand. Bom at Brussels, Oct. 14, 
1801 ; died at Ghent, Sept. 15, 1883. Professor of physics at 

Plato. Born at Athens. — 429; died in — 348. Founder of the 
Academy. Contributed to the philosophy of mathematics. 

Plato of Tivoli. Lived 1120. Translated Al Battani's trigonom- 
etry and other works. 

PlUcker, Johann. Bora at Elberfeld, July i6, 1801 ; died at Bonn. 
May 22, 1868. Professor of mathematics at Bonn and Halle. 
One of the foremost geometers of the century. 


Poisson, Simeon Denis. Bom at Pithiviera, Loiret. 1781 ; died 
at Paris, 1840. Chiefly known as a physicist. Contributed 
to the study of definite integrals and of series. 

Poncelet, Jean Victor. Born at Metz, 1788 ; died at Paris, 1867. 
One of the founders of projective geometry. 

Pothenot, Laurent. Died at Paris in 1732. Professor of mathe- 
matics in the College Royale de France. 

Proclus. Born at Byzantium, 412 ; died in 485. Wrote a com 
mentary on Euclid. Studied higher plane curves. 

PtoUmy (Piolem2ievLS Claudius). Born at Ptolemais, 87; died at 
Alexandria, 165. One of the greatest Greek astronomers. 

Pythagoras, Born at Samos, — 580 ; died at Megapontum, — 501. 
Studied in Egypt and the East. Founded the Pythagorean 
school at Croton, Southern Italy. Beginning of the theory of 
numbers. Celebrated geometrician. 

Quetelett Lambert Adolph Jacques. Born at Ghent, Feb. 22, 
1796; died at Brussels, Feb. 7, 1874. Director of the royal 
observatory of Belgium. Contributed to geometry, astronomy, 
and statistics. 

PamuSt Peter (Pierre de la Ram£e). Bom at Cuth, Ficardy, 1515 ; 
murdered at the massacre of St. Bartholomew, Paris, August 
24-25, 1572. Philosopher, but also a prominent writer on 

Pecorde, Robert. Bom at Tenby. Wales, 1510 ; died in prison, 
at London, 1558. Professor of mathematics and rhetoric at 
Oxford. Introduced the sign = for equality. 

Regiomontanus. Johannes Mtiller. Born near K(^nigsberg|June 
6, 1436 ; died at Rome, July 6, 1476. Mathematician, astron- 
omer, geographer. Translator of Greek mathematics. Author 
of first text- book of trigonometry. 

Remigius of Auxerre. Died about 908. Pupil of Alcuin's. Wrote 
on arithmetic. 

Rhaeticus, Georg Joachim. Bom at Feldkirch, 15 14 ; died at 
Kaschau, 1576. Professor of mathemntics at Wittenberg ; pu- 
pil of Copernicus and editor of his works. Contributed to 


Riccati, Count Jacopo Francesco. Born at Venice, 1676 ; died at 
Treves, 1754. Contribnted to physics and differe ntial equa- 

Richelot, Friedrich Julius. Bom at KOnigsberg, Nov. 6, 1808 ; 
died March 31, 1875 in same place. Wrote on elliptic and 
Abelian functions. 

Rtemann, George Friedrich Bemhard. Bom at Breselenz, Sept. 
17, 1826 ; died at Selasca, July 20, 1866. Contributed to the 
theory of functions and to the study of surfaces. 

Riese, Adam. Bom at Staffelstein, near Lichtenfels, 1492 ; died 
at Annaberg, 1559. Most influential teacher of and writer on 
arithmetic in the i6th century. 

Roberval, Giles Persone de. Born at Roberval, 1602 ; died at 
Paris, 1675. Professor of mathematics at Paris. Geometry 
of tangents and the cycloid. 

RoUe, Michel. Born at Ambert, April 22, 1652 ; died at Paris, 
Nov. 8, 1719. Discovered the theorem which bears his name, 
in the theory of equations. 

Rudolff, Christoff. Lived in first part of the sixteenth century. 
German algebraist. 

Sctcro-Bosco, Johannes de. Born at Holy wood (Halifax), York- 
shire. i20o(?); died at Paris, 1256. Professor of mathematics 
and astronomy at Paris. Chiefly known for his Tractatus de 
s^hcera mundi. 

Saint' Venant, Adh^mar Jean Claude Barr^ de. Bom in 1797 ; 
died in Venddme, 1886. Writer on elasticity and torsion. 

Saint' Vincent, Gregoire de. Bom at Bruges, 1584 ; died at Ghent. 
1667. Known for his vain attempts at circle squaring. 

Saurin, Joseph. Born at Courtaison, 1659; died at Paris, 1737. 
Geometry of tangents. 

Scheeffer, Ludwig. Born at K5nigsberg, 1859 ; died at Munich, 
1885. Writer on theory of functions. 

Schindel, Johannes. See Joannes de Praga. 

Schwenter, Daniel. Bom at Nuremberg, 1585 ; died in 1636. 
Professor of oriental languages and of mathematics at Altdorf . 

Serenus of Antissa. Lived about 350 Geometer. 


Serret, Joseph Alfred. Born at Paris, Aug. 30, 1819 ; died at 
Versailles, March 2, 1885. Author of well-known text-books 
on algebra and the differential and integral calculus. 

Sextus Julius Africanus. Lived about 220. Wrote on the his- 
tory of mathematics. 

Simfson, Thomas. Born at Bosworth, Aug. 20, 17 10; died at 
Woolwich, May 14, 1761. Author of text-books on algebra, 
geometry, trigonometry, and fluxions. 

Sluze, Ren^ Francois Walter de. Born at Vis^ on the Maas, 1622 ; 
died at Liege in 1685. Contributed to the notation of the cal- 
culus, and to geometry. 

Smith, Henry John Stephen. Born at Dublin, 1826 ; died at Ox- 
ford, Feb. 9, 1883. Leading English writer on theory of num- 

Snell, Willebrord, van Roijen. Bom at Leyden, 1581 ; died there, 
1626. Physicist, astronomer, and contributor to trigonometry. 

S-pottiswoode, William. Born in London, Jan. 11, 1825 ; died 
there, June 27, 1883. President of the Royal Society. Writer 
on algebra and geometry. 

Staudt, Karl Georg Christian von. Bom at Rothenburg a. d. 
Tauber, Jan. 24, 1798 ; died at Erlangen, June i, 1867. Prom- 
inent contributor to modem geometry, Geometrie der Lage. 

Steiner, Jacob. Born at Utzendorf, March 18, 1796 ; died at 
Bern, April i, 1863. Famous geometrician. 

Stevin, Simon. Born at Bruges, 1548 ; died at Leyden (or the 
Hague), 1620. Physicist and arithmetician. 

Stewart, Matthew. Born at Rothsay, Isle of Bute, 1717; died at 
Edinburgh, 1785. Succeeded Maclaurin as professor of math- 
ematics at Edinburgh. Contributed to modern elementary 

Stifel, Michael. Born at Esslingen, i486 or 1487; died at Jena, 
1567. Chiefly known for his Arithmetica integra (1544). 

Sturm, Jacques Charles Fran9ois. Born in Geneva, 1803 ; died 
in 1855. Professor in the fecole Poly technique at Paris. 
" Sturm's theorem," 

Sylvester, James Joseph. Born in London, Sept. 3, 1814 ; died 
in same place, March 15, 1897. Savilian professor of pure 


geometry in the University of Oxford. Writer on algebra, 
especially the theory of invariants and covariants. 

TabU tbn Kurra, Born at Harran in Mesopot'amia, 833 ; died at 
Bagdad, 902. Mathematician and astronomer. Translated 
works of the Greek mathematicians, and wrote on the theory 
of numbers. 

Tartaglia, Nicolo. (Nicholas the Stammerer. Real name, Ni- 
cole Fontana.) Born at Brescia, c. 1500; died at Venice, c, 
1557. Physicist and arithmetician ; best known for his work 
on cubic equations. 

Taylor, Brook. Born at Edmonton, 1685 ; died at London, 1731. 
Physicist and mathematician. Known chiefly for his work in 

Tholes, Born at Miletus, — 640; died at Athens, —548. One of 
the " seven wise men " of Greece ; founded the Ionian School. 
Traveled in Egypt and there learned astronomy and geom- 
etry. First scientific geometry in Greece. 

Theaetetus of Heraclea. Lived in — 390. Pupil of Socrates. 
Wrote on irrational numbers and on geometry. 

Theodorus of Cyrene. Lived in — 410. Plato's mathematical 
teacher. Wrote on irrational numbers. 

Theon of Alexandria. Lived in 370. Teacher at Alexandria. 
Edited works of Greek mathematicians. 

Theon of Smyrna Lived in 130. Platonic philosopher. Wrote 
on arithmetic, geometry, mathematical history, and astronomy. 

Thymaridas of Paros. Lived in — 390. Pythagorean ; wrote on 
arithmetic and equations. 

Torricelli, Evangelista. Born at FaSnza, 1608 ; died in 1647. 
Famous physicist. 

Tortolint, Barnaba. Born at Rome, Nov. 19, 1808 ; died August 
24, 1874. Editor of the Annali which bear his name. * 

Trenibley, Jean. Born at Geneva, 1749; died in 1811. Wrote 
on differential equations. 

Tschtrnhausen, Ehrenfried Walter, Graf von. Born at Kiess- 
lingswalde, 1651; died at Dresden, 1708. Founded the theory 
of catacaustics. 


UbcUdi, Guido. See Del Monte. 

Unger, Ephraim Solomon. Born at Coswig, 1788 ; died in 1870. 

Ursinus, Benjamin. 1587—1633. Wrote on trigonometry and 
computed tables. 

Van Ceulen, Ludolph. Born at Hildesheim, Jan. 18 (or 28), 1540 ; 
died in Holland, Dec. 31, 1610. Known for his computations 

of TT. 

Vandermonde, Charles Auguste. Bom at Paris, in 1735 ; died 
there, 1796. Director of the Conservatoire pour les arts et 


Van Eyckt Jan. 1385-1440. Dutch painter. 

Van Schooten, Franciscus (the younger). Born in 1615 ; died in 
1660. Editor of Descartes and Vieta. 

Viite (Vieta), Franjois, Seigneur de la Bigotiere. Born at Fonte- 
nay-le-Comte, 1540; died at Paris, 1603. The foremost alge- 
braist of his time. Also wrote on trigonometry and geometry. 

Vincent. See Saint-Vincent. 

VitruiHus. Marcus Vitruvius Pollio. Lived in — 15. Roman archi- 
tect. Wrote upon applied mathematics. 

Vnnantt Vincenzo. Bom at Florence, 1622 ; died there, 1703. 
Pupil of Galileo and Torricelli. Contributed to elementary 

Wallace, William. Born in 1768; died in 1843. Professor of 
mathematics at Edinburgh. 

WaJHs^ John. Born at Ashford, 161 6 ; died at Oxford, 1703. Sa- 
vilian professor of geometry at Oxford. Published many 
mathematical works. Suggested (1685) the modern graphic 
interpretation of the imaginary. 

Weierstrass, Karl Theodor Wilhelm. Born at Ostenfelde, Oct. 
31, 1815 ; died at Berlin, Feb. 19, 1897. O^*® of the ablest 
mathematicians of the century. 

Werner, Johann. Born at Nuremberg, 1468; died in 1528. Wrote 
on mathematics, geography, and astronomy. 

Widmann, Johann, von Eger. Lived in 1489. Lectured on alge- 
bra at Leipzig. The originator of German algebra. Wrote 
also on arithmetic and geometry. 


IVttt, Jan de. Bom in 1625, died in 1672 Friend and helper of 

IVolf, Johann Christian von. Bom at Breslau, 1679 ; died at 
Halle, 1754. Professor of mathematics and physics at Halle. 
and Marburg. Text-book writer. 

Woefcke, Franz. Bom at Dessau, May 6, 1826 ; died at Paris, 
March 25, 1864. Studied the history of the development of 
mathematical sciences among the Arabs. 

Wren-i Sir Christopher. Born at East Knoyle, 1632 ; died at Lon- 
don, in 1723. Professor of astronomy at Gresham College ; 
Savilian professor at Oxford : president of the Royal Society. 
Known, however, entirely for his great work as an architect 




\bacist8, 39, 41. 
Abacus, 15. 25, 26, 37. 
Abel, 62, 154, 155. 163. 181-188, 
r Abscissa, 229. 
, Abul Wafal, 225, 286. 
Academies founded, 1 16. 
Adelard (iCthelhard) of Bath, 74, 218. 
Africanus, S. Jul., 202. 
Ahmes, 19, 31. 33i 34t 11^ 78, i92i 28a. 
Alcuin, 41. 

Al Banna, Ibn, 30, 76, 9a 
Al Battani, 285. 
Albert! , 227. 
Algebra, 61, 7J^ 96, 107; etymology, 

88; first German work, iia 
Algorism, 75. 

Al Kalsadi, 30, 31. 75. 76, 89, 90, 92. 
Al Karkhi, 75, 93- 
Al Khojandi, 76. 
Al Khowarazmi, 29, 33. 74* 75i 88, 89, 

91, 217. 
Al Kuhi, 217. 
Alligation, 34. 
Almagest, 283. 
Al Nasawi, 30, 34. 
Al Sagani, 217. 
Amicable numbers, 35. 
Anaxagoras. 195, 213. 
Angle, trisection of, 196, 197, 207, 208, 

Annuities, 56, 148. 
Anton, I79«. 
Apian, 108, 288, 289. 
Apices, 15, 27. 37. 39- 
Apollpnius, 80, 152, 190, 200-209, 228, 

229, 231. 
Approximations in square root, 70. 

Arabs, 3, 15. 20. 35, 39, 53, 74, 76, 88, 

89, 191, 214, 285. 
Arbitration of exchange, 55. 
Arcerianus, Codex, 214, 2x8. 
Archimedes, 68-71, 78, 81-83, 190, 199, 

204, 205, 208, 210, 212. 
Archytas, 69, 82, 204, 207, 211. 
Argand, 124, 123. 
Aristophanes, 25. 
Aristotle, 64, 70. 

Arithmetic. 18, 24, 36. 49, 51, 64, 95. 
Aritlimetic, foundations of, 189; re- 

quired, 43. 
Arithmetical triangle, 1x8. 
Aronhold, 146, 250. 
Aryabhatta, 12, 72, 74, 215, 216. 
Aryans, 12. 
Associative law, 119. 
Assurance, 56-60. 
Astronomy, 18. 
August, 246. 
Ausdehnungslehre, 127. 
Austrian subtraction, 28, 48. 
Avicenna, 76. 
Axioms, 197. 

Babylonians, 9, 10, 14, 19, 24, 25, 63, 

64, 190, 192, 193. 
Bachet, 106, 134, 137. 
Ball, W. W. R.. i72». 
Baltzer, i67», 224». 
Bamberger arithmetic, 51. 
Banna. See Ibn al Banna. 
Bardin, 277. 
Barrow, 169, 238. 
Bartl, 167. 
Barycentrischer CalcUl^ 129, 330. 

*The numbers refer to pages, the small italic »'s to footnotes. 


Bede, je, s7, 10. 
BalliTiii*. i]0. 166. 
Ballniml, 14S, 169, 1; 

Bemonlli famllj. jS ; Jacob, 14B. I' 
151,171. '?]. 178.179. "39. "39; )ol 
IJJ. IM. 173. 17s, 17*. 179. "38. » 
i«S; Danial. 166, 17J. 

lieienl ds Haan. 

lindsr, 157. 
Ilnomial coefGcit 

BoaolakDwiky, ijg. 
BoDvallBi, 117. 
Boyi, 166. 

Briouhi. us, 1 
Brourd, US- 

Brane, sg. 

178; ot loitii:. 131; of Tn 



Cantor. G., lao. .13; Cuitor 

M,. jn. 

Capelli, T9J. 

Cardan, .01-103, 109. ,«, 111, 

'W 15s, 


Cardioid. m. 

Camot. 174, »«, 1*6. 14S. 

CaialDi'8 0T,l.2... 


Cauls problem of Arohimed 


Canchy, 6a. 119, 114 „j, ijfl. 

■19, MS, 

1)3, i!4, 164. .67. 168, .7,, 

81, ISB, 


Canslics, 138, 

«34- 133, 

CajlEj. «6«., iig«.. .a,. ,4j, 

46, 16S 

178, 553. 1S7, afi3. "6*. 266, i; 


Ceva. J44. 

Chain rule, s»,js. 

Cliance. See Probabiliiiei, 

Chappie, a«4. 



Cloister schools. See Charch schools. 
Codex Arcerianus, 214. 
Coefficients and roots, 115, 156. 

Cohen, xyzn. 

Cole, i62n. 

Combinations, 70, 74, 150, 151. 

Commercial arithmetic, 22, 51, 60. 

Commutative law, 119. 

Compasses, single opening, 225. 

Complementary division, 38. 

Complex numbers, 73, joi, 123, 126, 
182. Complex variable. See Func- 
tions, theory of. 

Complexes, 254. 

Compound interest, 53* 

Computus, 37, 39. 

Conchoid, 211. 

Condorcex U9> 

Conies, 8z, 202, 204-208, 228, 230, 239, 

Congruences, theory of, 131. 

Conon, 2ZO. 

Conrad, H., 109. 

Conrad of Megenberg, 319. 

Contact transformations, 178, 269, 276. 

Continued fractions, 131-133, 168. 

Convergency, 152-155, 189. See Se- 

Coordinates, Cartesian, 231 ; curvi- 
linear, 268, 269; elliptic, 269. 

Copernicns, 289. 

Correspondence, one-to-one, 251,264, 
266, 263. 

Cosine, 288. 

C08S, 96-99, 107, X09, III. 

Cotes, 174, 239. 241. a44. 

Connting, 6. 

Cousin, 227. 

Covariants, 146. See also Forms, In- 

Cramer, 132. 167, 240; paradox, 240. 

Crelle, 141, 245, 257. 

Cremona, 256, 266. 

Crofton, 276. 

Cross ratio, 258, 259. 

Cube, duplication of, 82, 104, 204, 207: 
mnltipli cation of, 207, 2zi. 

Culvasutras, 72. 

Cuneiform inscriptions, 9. 

Cunynghame, 166. 

Curtse, zSgn. 

Cnrratnre, measore of, 368. 

Curves, classification of, 333, 239> ^0; 
deficiency of, 262, 263 ; gauche (of 
double curvature), 243, 255* 9lBi%; 
with higher lingulmrities, 253. 

Cusa, 237. 

Cycloid, 178, 337f «S8* 

</, symbol of difiEerentiation. 170-172, 

3, symbol of differentiation, 180. 

D'Alembert, 175, z8o. 

Dante, 94. 

DeBeaune, 156. 

Decimal fractions, 50. 

Decker, 292. 

Dedekind, 120-122, 126, T27, 189. 

Defective nnmbers, 35. 

Deficiency of curves, 262. 263. 

Definite integrals, 174. 

Degrees (circle), 34. 

De Lagny, 157. 

De la Goumerie, 361. 

Delambre, 295. 

De r Hospital, 173, 178, 179. 

Delian problem, 82, 104, 204, 207. 

Democritns, 213. 

De Moivre, 124, 152, 160. 

De Morgan, 143, 155. 

Desargues, 205, 337, 242, 259. 

Descartes, 4, loB, 117. 119, 124, 136, 

140, 156, 191, 228, 230-233, 238. 
Descriptive geometry, 247, 259, 260. 
Determinants, 133, 144, Z45, 167, 168, 

DeWitt, 57, 148. 
Dialytic method, 144, 145. 
Diametral numbers, 105. 
Differential calculus, 168, 170, 171,178; 

equations, 174-178, 269; geometry, 

Dimensions, «., 375. 
Dini, 155, 189. 
Dinostratus, I97t 2ia 
Diodes, 211. 
Diophantus, 65, 70, 77, 81, 84, 85, 90, 

93. 133. 134. 
Dirichlet, 62, 125. 126, 133, 139, 140, 

153. 174. 177. 181, 189, 279. 
Discount, 54. 
Discriminant, 145. 
Distributive law, 119, 130. 




of 111 

Dyck. tja. 

I, InaUoaaliir a(, tj]. 

Eanar, 41. 

Bcole polyUchoique. aCt, 



7. 13B. 

lis, '35-160, 1 64-166; 
95, B4, B6, 93. 133, 
77.78, 87. 90; limii 

ralie,79-Si,gj,9i, i, 

Euclid, jj, 65-fi», 79, «o, I 

PsKnaiio, 180, iSl. 


Felkel. 141. 

Fermal. 57, u8, ij«. ijj, ,j?. ,^ y. 

16S, I7J. 129, 134. 
pBirari, III, ijj, Mj. 
Pecra, 113. 

PibODBCCi S« LMurdo. 

FiDcIi, 1S8. 

Pingar raekonlng, 25, j6, 43. 

Fischer, », 

Pluilon*. 171, r73. 

Fonns, Iheory ot, 131, 14J-147. 

Fonotiona, Absliui, iSo, 186, iSS. 189. 
elliptic. [$]. iSo-tgi; periodicity ot, 
184; syiDinelrio, 141,141; Iheotj gt' 
177, 180, tSi. 1B8: then, iSi, 1S8, iS» 

PundHmenial lava of muaber. iig 

I6r, tra. tji. ai 
Geber, iiS6. 
Gslli brand «!. 

Ganoccbl, isg- 
GeomelTic meu 


Gelbart, i;, 37, to, 61, «8. 
Gergoooe, 149. ij?- 
Gaihud ol CrBmoni. 41), 18G. 
Gerbardt, 47". 
Ueroua algebra. 96. 107; univa 

Gleiiiig, ia6H. 

Gltla- «chHli, II. 
Gliab, 9. 
Glaisher, 14I- 

Golden Kciion, igj. » 
GotdM, 144, 146, 147. 
Goumarle. 161. 

Haho, 48' 

Hallar, 37. : 
Halpben, M 

Heron, 64, 70,7s, 81. 84. ».. 1 
Messa, 143--14}, iGt. 16B, 176. 3 

Hindn algebra, 84; srilhmplic. 

s, 6},8k 197, 104,113. 

Homology. J49. 
Hoppe, 167. :73-.,»45. 
Hoapllal. 173. 178, I7». 

Hujgens, 131, 148, IM, 1)1, ■«■. 
Hyperbola, Si, to). 
Hypeibolold, 141. 



Hyperdeterminants, 146. 
HyperelUptic integrals, 187. 
Hypergeometric series, 153. 
Hypsicles, 84. 200. txt. 

/ lor 1^—1, i«4. 

lamblichus, 136. 

Ibn al Banna, 30, 76, 90. 

Ibn Knrra, 136, 217. 

Icosahedron theory, x66. 

Ideal numbers, 126. 

Imaginaries. See Complex numbers. 

Incommensurable quantities. 69. 

Indeterminate equations. See Equa- 

Indivisibles, 234, 236 

Infinite, 173. See Series. 

Infinitesimals, 169, 170, 173, 174. 

Insertions, 208, 211. 

Insurance, 56-58. 

Integral calculus, 174, 178. 

Interest, 54. 

Invariants, 145-148, 262, 274. 

Involutes, 238, 241. 

Involutions, 252. 

Irrational numbers, 68, 69, 100, 119, 
122, 123, 133, 189. 

Irreducible case of cubics, 112. 

Isidorus, 36. 

Isoperimetric problems, 179. 200 

Italian algeora, 90. 

Jacobi, 62, 138, 139. 143. 144. 165, 168, 
174-177. 181-187, 269, 276, 279. 

Johann von Gmunden, 95. 

Jonquiferes, 256. 

Jordan, 165. 

Kalsadi. See AI Kalsadi. 

Karup, 56, 59. 

KHstner, 48. 

Kepler, 4, 50, 61, 169, 173, 191, 222-224, 

24s, 288. 
Khayyam. 75, 89, 92, 93. 
Khojandi, 76. 

Khowarazmi. See Al Khowarazmi. 
Klein, 147, 165, 177, 178, 207*. , 254, 274, 

277, 278. 
Knilling, 23. 
KOnigsberger, x8o. 
Kossak, X20M. 

Krafft, 135. 

Kronecker, 139, 165 

Krfiger, 141. 

Krunibiegel and Amthor, 83. 

Kummer, 126, 138, i39«.. 155, 270, 278. 

Kurra, Tabit ibn, 136, 217." ' 

Lacroix, 242, 261. 

Lagny, De, 157. 

Lagrange, 62, 136, 138, 143, 151, 159 
160, 166, 167, 173, 175, 176, 179, 180, 
182, 239, 267, 294* a95. 

Laguerre, 274. 

Labi re. 106, 249. 

Lalanne, 167. 

Laloub6re, 158. 

Lambert, 124, 133, 141, 260, 267, 295. 

Lam4, 240, 269. 

Landen, 180, 182, 244. 

Lansberg, 249. 

Laplace, 150, 151, 167, 175. 

Latin schools, 21, 43. 

Least squares, 149. 

Lebesgne, 139. 

Legendre, 133, 136, 138-140, 149, 166, 
174. 180-184, 187, 270, 295. 

Lehmus, 257. 

Leibnitz, 4, 48, 54, 58, 62, 117. 150-152, 
156. 167, 170-173, 178, 229, 239, 242. 

Lemniscate, 241. 

Lencker, 227. 

Leonardo da Vinci. 225; of Pisa (Fi- 
bonacci), 40, 41, 45. 95, 101, 107. 109, 
111, 218. 

Leseur, 158. 

Lessing, 83. 

Letters used for quantities, 64. 

Lexell, 295. 

L' Hospital, 173, 178, lyg, 

Lhuilier, 244. 

Lie, 147, 177, 269, 276. 

Lieber, 245». 

Light, theory of, 270. 

Limapon, 241. 

Limits of roots, 156. 160, 166. 

Lindemann 133, 189, 207. 

Liouville, 139, 181, 269. 

Lipschitz. 147. 

Lituus. 241. 

Lobachevsky, 271. 

Loci, 209, 210, 232. 

LosHrlihmlc uilei, iji: 
Lc^e, calcului of, tji. 

MacUarin, 15a. 136, i: 

Menalmns, 1B3. 

lObius, iiS, 129, liZ, «4. >49> iio-'l 
lodalB, goouieuic, I7«, 
lotunuaadnas. 3. See Araba. 

le. 106. 

[76, 176, 247, 148, a67. »7T- 


iDltipUcailan, 4J. 4& 
[jHic haiagram. IJ7, " 

.no. C, 269; K., 17- 
n. 4. 6», "7-"'9. W. 
75, 178. 2J4, »3»- 

iclideaa gaometry, ^ft 

■ra, amicable, 136; clasaea of, 
on<:aplo(. iS. .2<>;ldaal,i^l 

e, 67. M, 136, 141, 161. 162; p7- 
dil. 711 plana and aoUd. G6; 
una of, 6; Ihsorr of, i33-'«*- 


OrB«mB, gi.ioa.iat. 

Potnoart, iSj, 177. 



Onihited, 117, i^ 

Point RiOBpa. 14ft 

Polar. 149. as*. 

w. nuun of. I]], ID? : valnes of. >»■, 

Pole, 149. 


Facloli. 41. 45-47, 3* 9S. 9*. i<". 

Polygons, aWr, .18, i«, =4. 


Pooctiot. 346, 148, 549. «!». >58. »«S- 


Parabola. Bi; araa,«8; Dame, loj. 


Paraboloid, 141 

Power series. .03 

Prime nnmbers, 67. fiB, 156, mi. i8> 

Panidoii of perigon i«o-lBl. 

Paacal. 4S. ij. ..fl, 148, ijo, 159, .75, 

ProbabiUties, 148. 14a, <}«. 



Projection, ai], 114. SeaCaonatr;. 

P«nl<«r, iss. itti. 

Proportion, 7B. 109, 

Pair«, HI. 

Ptolemr, 201, ai4, tX. *). 

PsIsUar, III. 

Puiilea, 54, 


Pythagoras, 68. 179. 190, t9(. igj, 114 

Pepin, IM- 

PerfKt namben, ». SB. 

IBS, 198. 

Periodicity of f unciiooi. iS|. 

Perspective, aifi. 117. 159. 

Petty, J7- 

h, J, 41, 4J, loj 


IH. ITS, 17«- 



Phoeniciwu. 8, ra. 





PUio, 67 

81.197,107: of 


Bodiea. t«. 

Pliny, M 


44. 1»- M9-aS* 



H6J, i7I.aW. 


equations. Vi 

Pltls. S 




1, 1*. M. 90. 

ReguUr poljRoni 
«J. n6. 537. 145 
Rdff. IS.... irSn. 

Resiitlanl. if 1-143 
Reuschle. 14?. 167 
Reymen, 9S. 9B, f 
Rhabda, ij. 
RbaeliciiB. iSB. 

ScherlDg, 1S9. 

SchBnb.1,98, ,11 
Sch legal, 117* , 1 
SchlealngsT. i7j> 
SchoDUn. Vu. 1 
ScholUir. tS9. 
Sehi6der. Ijl. 
Scbaberl, 346, ib 

Schwenlcr, 131, 1 
Sciplone del FeT 

Secant, 288. 

, laj. no, 141. *W 



S.idel. IH. 


Sedea, «. (>7. ri. 74. 70. 

. >9, -14. 


.56; cube, 

Serret, .65. 14«-. 

al and Im- 

Swvoia, M9. 


«. 73. lOJ- 


milb. D. G . 178a. See Bou 

Sutell. »77. 

Solid Domben, 66 

Sp-in, 1. 

Spirals, 141 lolArcb 

m^«, «B. 

Sc|,.^re.. =flE., ,,9. 

Squatinn circle. Se 


StabI, ,89. 

Star polygons, aiB, I 

9, "4. 

Sleiner, 225, 246. 249, 




Stewart, 244. 

Stifel. 4, 49, 52, 53, 97. 99-105, 109-111, 
itSt "5. "8, 220, 221, 224. 

Stokes, 1S4. 

Stoll, 246. 

Stolz, i9om'. 

Stringham, 243. 

Stubbs, t66. 

Sturm, 48, 270. 

Substitutions, groups, 164, 165. 

Sun tse, 87. 

Surfaces, families of, 267; models of, 
877 • of negative curvature, 273 ; 
second order, 213, 262 ; third order, 
963 ; skew, ass ; Steiner, 256; ruled, 

Surveying, 18, 71. 

Suter, 94ii. 

Swan pan, flS. 

Sylow, i6s. 

Sylvester, 143-147, 97s. 

Sylvester II., Pope, is. 

Symbols, 47, 63, 6s, 71. 76, 88, 89, 9$- 
97. 99. 102, X08, 109, X17. 170, 171. 183, 

Symmedians, 34s. 

Symmetric determinants, 168; func- 
tions, 142, 143. 

Tabit ibn Kurra, 136, 217. 

Tablet, astronomical, 286; chords, 
2S2; factor, 141; mortality, 148; 
primes, 141 ; symmetric functions, 
X43 ; sines, 286 ; theory of numbers, 
142; trigonometric, 282, 286, 289, 
290, 293. 

Tacquet, 174. 

Tanck, 23. 

Tangent, 288. 

Tannery, 33, 70, 120. 

Tartaglia, 3, 49, 51. S2. "«. "S. i55. 

Tatstha, 29. 

Taylor, B., 152, i66, 259; C, 224«. 

Thales, 194. 

Theaetetus, 212. 

Theodorus, 69 

Theon of Alexandria, 34, 70 

Thieme, 244. 

Thirty years' war, 22. 

Thom6, 177. 

Thompson, 107, 266. 

Timaeus, 3x2. 

Tonti, 56. 

Tontines, S7* 

Torricelli, 337. 

Torus, 2x3. 

Transformations of contact, 178, 269 

Transon, 270. * 

Transversals, 144, 248. 
Trenchant, 47. 
Treutlein, S2»., 67. g6n., gyn 
Trigonometry, 281. 
Tri section. See Angle. 
Trivium, 94. 
Tschirnhausen, 157, iS9, 165, 178, 238, 

Mh 34a. 
Tylor, 6m. 

Ubaldi, 338. 
Ulpian, 56. 
Unger, i6». 

Universities, rise of, 94. 
Unverzagt, X29«., X30». 

Valentiner, 256. 

Van Ceulen, 222. 

Van der Eycke, 222. 

Vandermonde, 118, 159, 167. 

Van Eyck, 226. 

Van Schooten, 136, 141, 156, 242. 

Variations. See Calculus. 

Vector, 130. 

Vedas, 25. 

Veronese, 373. 

Versed sine, 288. 

Victorius, 27. 

Vieta, 107, 108, 115, 117, 119, i34i 156, 

191, 222, 229, 249, 287, 288. 
Vincent, St., 151. 
Vitruvius, 215. 
VI acq, 292. 
Voigt, 139. 

Von Staudt, 162, 246, 249, 257-259. 263. 
Vooght, 292. 

Wafa, 825, 286. 

Wallis, 117, 125, 131, 135, 154, 173, 234. 

236, 237, 242. 
Waring, 143, 159, 239. 



Weber, 189. 

Weierstrass, 62, 190, 147, 178, x8x, 189. 

Welsh counting, 8 ; practice, 53. 

Wessel, 125. 

Widmann, 47, 51, aao. 

Wiener, 226»., 245, 278. 

Witt. De, 57, 148. 

Witts tein, 59, 256H. 

Wolf, 47. 48. 

Woodhouse, i78«. 

Woolhousa, ^Gm. 

Wordsworth, 19m, 
Wren, 243, 247- 

X, the symbol, 97. 
Year, length of, 34. 

Zangemeister, xi. 
Zeller, 139. 
Zenodorns, 200. 
Zero, 12, 16, 39, 40, 74. 
Zeuthen, 68»., 253, 264 





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